Index
stringlengths 1
5
| Challenge
stringlengths 41
1.55k
| Answer in Latex
stringclasses 122
values | Answer in Sympy
stringlengths 1
774
| Variation
stringclasses 31
values | Source
stringclasses 61
values |
|---|---|---|---|---|---|
201
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = e^{ \left(649916793\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(649916793 - 10*649916793**3 + 649916793**5)/120 + x**4*(-4*649916793**2 + 649916793**4)/24 + x**3*(-649916793 + 649916793**3)/6 + x**2*649916793**2/2 + x*649916793 + 1
|
Numeric-One-9
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
202
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = e^{ \left(9135588304\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(9135588304 - 10*9135588304**3 + 9135588304**5)/120 + x**4*(-4*9135588304**2 + 9135588304**4)/24 + x**3*(-9135588304 + 9135588304**3)/6 + x**2*9135588304**2/2 + x*9135588304 + 1
|
Numeric-One-10
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
203
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(9\right) e^{ \left(6\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(6 - 10*6**3 + 6**5)*9/120 + x**4*(-4*6**2 + 6**4)*9/24 + x**3*(-6 + 6**3)*9/6 + x**2*6**2*9/2 + x*6*9 + 9
|
Numeric-All-1
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
204
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(98\right) e^{ \left(55\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(55 - 10*55**3 + 55**5)*98/120 + x**4*(-4*55**2 + 55**4)*98/24 + x**3*(-55 + 55**3)*98/6 + x**2*55**2*98/2 + x*55*98 + 98
|
Numeric-All-2
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
205
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(286\right) e^{ \left(426\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(426 - 10*426**3 + 426**5)*286/120 + x**4*(-4*426**2 + 426**4)*286/24 + x**3*(-426 + 426**3)*286/6 + x**2*426**2*286/2 + x*426*286 + 286
|
Numeric-All-3
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
206
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(6613\right) e^{ \left(3668\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(3668 - 10*3668**3 + 3668**5)*6613/120 + x**4*(-4*3668**2 + 3668**4)*6613/24 + x**3*(-3668 + 3668**3)*6613/6 + x**2*3668**2*6613/2 + x*3668*6613 + 6613
|
Numeric-All-4
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
207
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(54737\right) e^{ \left(79738\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(79738 - 10*79738**3 + 79738**5)*54737/120 + x**4*(-4*79738**2 + 79738**4)*54737/24 + x**3*(-79738 + 79738**3)*54737/6 + x**2*79738**2*54737/2 + x*79738*54737 + 54737
|
Numeric-All-5
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
208
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(714540\right) e^{ \left(928533\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(928533 - 10*928533**3 + 928533**5)*714540/120 + x**4*(-4*928533**2 + 928533**4)*714540/24 + x**3*(-928533 + 928533**3)*714540/6 + x**2*928533**2*714540/2 + x*928533*714540 + 714540
|
Numeric-All-6
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
209
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(5068281\right) e^{ \left(3537545\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(3537545 - 10*3537545**3 + 3537545**5)*5068281/120 + x**4*(-4*3537545**2 + 3537545**4)*5068281/24 + x**3*(-3537545 + 3537545**3)*5068281/6 + x**2*3537545**2*5068281/2 + x*3537545*5068281 + 5068281
|
Numeric-All-7
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
210
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(32593878\right) e^{ \left(20771359\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(20771359 - 10*20771359**3 + 20771359**5)*32593878/120 + x**4*(-4*20771359**2 + 20771359**4)*32593878/24 + x**3*(-20771359 + 20771359**3)*32593878/6 + x**2*20771359**2*32593878/2 + x*20771359*32593878 + 32593878
|
Numeric-All-8
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
211
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(546801591\right) e^{ \left(852933933\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(852933933 - 10*852933933**3 + 852933933**5)*546801591/120 + x**4*(-4*852933933**2 + 852933933**4)*546801591/24 + x**3*(-852933933 + 852933933**3)*546801591/6 + x**2*852933933**2*546801591/2 + x*852933933*546801591 + 546801591
|
Numeric-All-9
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
212
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = \left(6069995524\right) e^{ \left(8637358860\right) \sin(x)} $, where A and B are symbolic constants.
|
x**5*(8637358860 - 10*8637358860**3 + 8637358860**5)*6069995524/120 + x**4*(-4*8637358860**2 + 8637358860**4)*6069995524/24 + x**3*(-8637358860 + 8637358860**3)*6069995524/6 + x**2*8637358860**2*6069995524/2 + x*8637358860*6069995524 + 6069995524
|
Numeric-All-10
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
213
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = e^{B \sin(x)} $, where A and B are symbolic constants.
|
B**2*x**2/2 + B*x + x**5*(B**5 - 10*B**3 + B)/120 + x**4*(B**4 - 4*B**2)/24 + x**3*(B**3 - B)/6 + 1
|
Symbolic-1
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
214
|
Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) = A e^{ \sin(x)} $, where A and B are symbolic constants.
|
-A*x**5/15 - A*x**4/8 + A*x**2/2 + A*x + A
|
Symbolic-1
|
U-Math
sequences_series
1ccc052c-9604-4459-a752-98ebdf3e0764
|
|
215
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
216
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
217
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
218
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
219
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
220
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
221
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
222
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
223
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
224
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
225
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
226
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
227
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
228
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
229
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
230
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
231
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
232
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
233
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
234
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
235
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
236
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
237
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
238
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
239
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
240
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
241
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
242
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
243
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
244
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
245
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
246
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
247
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
248
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
249
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
250
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
251
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
252
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
253
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
254
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
255
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
256
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
257
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
258
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
259
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
260
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
261
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
262
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
263
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
264
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
265
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
266
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
267
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
268
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
269
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
270
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
271
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
272
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
273
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
274
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
275
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
276
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
277
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
278
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
279
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
280
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
281
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
282
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
283
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
284
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
285
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
286
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
287
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
288
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
289
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
290
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
291
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
292
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
293
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
294
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
295
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
296
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
297
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
298
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
299
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
300
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
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