<s>[INST] Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$ Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac {4x+53}{(x-6)(x-5)}\to\frac {4x+53}{(x-6)(x+5)}=\frac{A}{(x-6)}+\frac{B}{(x+5)}=\frac{7}{(x-6)}+\frac{-3}{(x+5)}=\frac{7}{-6(1-\frac x6)}+\frac{-3}{5\left(1-(-\frac x5)\right)}$
$\displaystyle=-\frac76\sum_{n=0}^\infty{\left(\frac x6\right)^n}-\frac35\sum_{n=0}^\infty(-1)^n\left(\frac x5\right)^n$
Now i'm having trouble with finding the radius of convergence and $f^{(20)}(0)$.
What does $f^{(20)}(0)$ means? 20 derivatives and $x=0$? That means i should find a some-kind of pattern and then place $x=0$?
 [/INST] When you have 
$$f(x) = \sum_{n=0}^{\infty} a_n x^n$$
then
$$f^{(k)}(0) = k! \, a_k$$
You know $a_{20}$:
$$a_{20} = -\frac{7}{6} \frac{1}{6^{20}} - \frac{3}{5} \frac{1}{5^{20}}$$
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