<s>[INST] Find Power Series representation of the function $f(x) = {x\over 2x^2 + 1}$? 
Find Power Series representation of the function $f(x) = \dfrac x{2x^2 + 1}$?

I'm not sure how to tackle this...I'm supposed to find interval of convergence. 
 [/INST] $\sum x^n = \frac {1}{1-x}$ when $x$ is in the radius of convergence
$\sum (-x)^n = \frac {1}{1+x}\\
\sum (-x^2)^n =\sum (-1)^nx^{2n} =  \frac {1}{1+x^2}\\
\sum (-1)^n(\sqrt 2 x)^{2n} =\sum (-1)^n(2^n) x^{2n}  \frac {1}{1+2x^2}\\
x\sum (-1)^n(2^n)x^{2n} = \frac {x}{1+2x^2}\\
\sum (-1)^n(2^n)x^{2n+1} = \frac {x}{1+2x^2}$
The series conveges by the root test if:
$\lim_\limits{n\to \infty}|\sqrt[n]{a_n} x|<1$
$|\sqrt[n]{a_n}| = |2^{\frac {n}{2n+1}} x|<1$
$|x|<\frac {\sqrt 2}{2}$
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