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
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|---|---|---|---|---|---|
301
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
302
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
303
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
304
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
305
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
306
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
307
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
308
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
309
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
310
|
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 $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
311
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
312
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ 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
|
|
313
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ 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
|
|
314
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ 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
|
|
315
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ 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
|
|
316
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ 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
|
|
317
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-All-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
318
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ 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
|
|
319
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ 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
|
|
320
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ 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
|
|
321
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ 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
|
|
322
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ 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
|
|
323
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ 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
|
|
324
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ 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
|
|
325
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ 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
|
|
326
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ 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
|
|
327
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ 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
|
|
328
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ 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
|
|
329
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ 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
|
|
330
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ 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
|
|
331
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ 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
|
|
332
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ 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
|
|
333
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ 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
|
|
334
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ 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
|
|
335
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- A x \right)} + \cos^{2}{\left(A x \right)}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
336
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(A - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(A - 1\right) \right)} + 1\right) \tan{\left(A x \right)}}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
337
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(A x \right)} + \cosh^{2}{\left(A x \right)}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
338
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(A x + \sqrt{A^{2} x^{2} + 1} \right)} \right)}}{A x}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
339
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(A)}{\ln(A)}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
340
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_A\left(\frac{x}{e}\right) + \log_A(e)}{\log_A(x)}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
341
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{2^{- N} x}{A}}{x}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
342
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(\frac{A \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} A}}{x}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
343
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i A x} - e^{- i A x}\right)}{2 \sin{\left(A x \right)}}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
344
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i A x} + 1\right) \tan{\left(A x \right)}}{\left(1 - e^{4 i A x}\right) \left(1 - \tan^{2}{\left(A x \right)}\right)}\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
345
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(3\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*3/6
|
Numeric-One-1
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
346
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(35\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*35/6
|
Numeric-One-2
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
347
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(789\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*789/6
|
Numeric-One-3
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
348
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(5806\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*5806/6
|
Numeric-One-4
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
349
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(12902\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*12902/6
|
Numeric-One-5
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
350
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(668529\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*668529/6
|
Numeric-One-6
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
351
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(5503043\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*5503043/6
|
Numeric-One-7
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
352
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(37840584\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*37840584/6
|
Numeric-One-8
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
353
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(426865087\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*426865087/6
|
Numeric-One-9
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
354
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(7755469684\right) n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*7755469684/6
|
Numeric-One-10
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
355
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\sin^{2}{\left(- B x \right)} + \cos^{2}{\left(B x \right)}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
356
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(B - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(B - 1\right) \right)} + 1\right) \tan{\left(B x \right)}}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
357
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \sinh^{2}{\left(B x \right)} + \cosh^{2}{\left(B x \right)}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
358
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\sinh{\left(\log{\left(B x + \sqrt{B^{2} x^{2} + 1} \right)} \right)}}{B x}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
359
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\ln(x) \cdot \log_{x}(B)}{\ln(B)}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
360
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{\log_B\left(\frac{x}{e}\right) + \log_B(e)}{\log_B(x)}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
361
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{2^{- N} x}{B}}{x}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
362
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(\frac{B \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} B}}{x}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
363
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{i \left(e^{i B x} - e^{- i B x}\right)}{2 \sin{\left(B x \right)}}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
364
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(- \frac{2 i \left(e^{4 i B x} + 1\right) \tan{\left(B x \right)}}{\left(1 - e^{4 i B x}\right) \left(1 - \tan^{2}{\left(B x \right)}\right)}\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
365
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(9\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*9/6
|
Numeric-One-1
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
366
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(79\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*79/6
|
Numeric-One-2
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
367
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(222\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*222/6
|
Numeric-One-3
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
368
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(6013\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*6013/6
|
Numeric-One-4
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
369
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(99455\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*99455/6
|
Numeric-One-5
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
370
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(370566\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*370566/6
|
Numeric-One-6
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
371
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(6925325\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*6925325/6
|
Numeric-One-7
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
372
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(69770590\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*69770590/6
|
Numeric-One-8
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
373
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(622741480\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*622741480/6
|
Numeric-One-9
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
374
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(5729172119\right) n \cdot ( x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))*5729172119/6
|
Numeric-One-10
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
375
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\sin^{2}{\left(- F x \right)} + \cos^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
376
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\tan{\left(x \right)} + \tan{\left(x \left(F - 1\right) \right)}}{\left(- \tan{\left(x \right)} \tan{\left(x \left(F - 1\right) \right)} + 1\right) \tan{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
377
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \sinh^{2}{\left(F x \right)} + \cosh^{2}{\left(F x \right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
378
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\sinh{\left(\log{\left(F x + \sqrt{F^{2} x^{2} + 1} \right)} \right)}}{F x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
379
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\ln(x) \cdot \log_{x}(F)}{\ln(F)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
380
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{\log_F\left(\frac{x}{e}\right) + \log_F(e)}{\log_F(x)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
381
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{2^{- N} x}{F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
382
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(\frac{F \sum_{N=1}^{\infty} \frac{6 x}{\pi^{2} N^{2} F}}{x}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
383
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \frac{i \left(e^{i F x} - e^{- i F x}\right)}{2 \sin{\left(F x \right)}}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Easy
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
384
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(- \frac{2 i \left(e^{4 i F x} + 1\right) \tan{\left(F x \right)}}{\left(1 - e^{4 i F x}\right) \left(1 - \tan^{2}{\left(F x \right)}\right)}\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1), (n, 2, oo))/6
|
Equivalence-One-Hard
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
385
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(2\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(2\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*2**n, (n, 2, oo))/6
|
Numeric-One-1
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
386
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(54\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(54\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*54**n, (n, 2, oo))/6
|
Numeric-One-2
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
387
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(867\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(867\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*867**n, (n, 2, oo))/6
|
Numeric-One-3
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
388
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(3801\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(3801\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*3801**n, (n, 2, oo))/6
|
Numeric-One-4
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
389
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(57767\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(57767\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*57767**n, (n, 2, oo))/6
|
Numeric-One-5
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
390
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(571517\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(571517\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*571517**n, (n, 2, oo))/6
|
Numeric-One-6
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
391
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(5409610\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(5409610\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*5409610**n, (n, 2, oo))/6
|
Numeric-One-7
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
392
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(56761193\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(56761193\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*56761193**n, (n, 2, oo))/6
|
Numeric-One-8
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
393
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(786200900\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(786200900\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*786200900**n, (n, 2, oo))/6
|
Numeric-One-9
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
394
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(8382186640\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( n \cdot ( \left(8382186640\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*8382186640**n, (n, 2, oo))/6
|
Numeric-One-10
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
395
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(5\right) n \cdot ( \left(8\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(7\right) n \cdot ( \left(8\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*8**n, (n, 2, oo))*7*5/6
|
Numeric-All-1
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
396
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(76\right) n \cdot ( \left(59\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(43\right) n \cdot ( \left(59\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*59**n, (n, 2, oo))*43*76/6
|
Numeric-All-2
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
397
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(769\right) n \cdot ( \left(763\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(297\right) n \cdot ( \left(763\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*763**n, (n, 2, oo))*297*769/6
|
Numeric-All-3
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
398
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(8937\right) n \cdot ( \left(5433\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(7365\right) n \cdot ( \left(5433\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*5433**n, (n, 2, oo))*7365*8937/6
|
Numeric-All-4
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
399
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(39530\right) n \cdot ( \left(68250\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(80927\right) n \cdot ( \left(68250\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*68250**n, (n, 2, oo))*80927*39530/6
|
Numeric-All-5
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
|
400
|
Find the (infinite) power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left( \left(329249\right) n \cdot ( \left(524768\right) x)^n\right)$ and $g(x) = \sum_{n=1}^\infty \left( \left(314629\right) n \cdot ( \left(524768\right) x)^n\right$
|
Sum(n*x**n*(n**2 - 1)*524768**n, (n, 2, oo))*314629*329249/6
|
Numeric-All-6
|
U-Math
sequences_series
fb6418ae-3440-4258-9388-89d799fd859a
|
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