幂级数好题
Example
\[
\sum_{n=0}^{\infty} \frac{(-2)^{n}+2}{2^{n}(2n+1)}x^{2n}
\]
直接泰勒走起！
\[
\begin{aligned}
\sum_{n=0}^{\infty} \frac{(-2)^{n}+2}{2^{n}(2n+1)}x^{2n} &= \frac{1}{x}\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1}x^{2n+1}+2\sum_{n=0}^{n} \frac{1}{2n+1}\left( \frac{x}{\sqrt{ 2 }} \right)^{2n}\\
&= \frac{\arctan x}{x} + \frac{\sqrt{ 2 }}{x} \cdot 2\sum_{n=0}^{\infty} \frac{1}{2n+1}\left( \frac{x}{\sqrt{ 2 }} \right)^{2n+1}\\
&= \frac{\arctan x}{x} + \frac{\sqrt{ 2 }}{x} \cdot \ln \frac{1+ \frac{x}{\sqrt{ 2 }}}{1- \frac{x}{\sqrt{ 2 }}} \\
&= \frac{\arctan x}{x} + \frac{\sqrt{ 2 }}{x} \ln\frac{\sqrt{ 2 }+x}{\sqrt{ 2 }-x}
\end{aligned}
\]
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