For series $\sum\limits_{n=1}^{\infty} a_nx^n$ where
$a_n = \{\begin{array}{lr}
1/n, & \text{n is even}\\
1/n^2, & \text{n is odd}\\
\end{array}$
Let $S_{e,n}$ be the $n^{th}$ partial sum of the series for even terms and $S_{o,n}$ be the $n^{th}$ partial sum of the series for odd terms. Prove both of these partial sums converge on (-1,1). Also, usnig the converge of these prove that the parital sums of the original series also converge on (-1,1) and so the series converges on (-1,1).
Trying to review and don't know what to do. Thanks for any help!
Best Answer
HINT:
Apply the ratio test (or root test) to the series $\sum_{n=1}^\infty \frac{x^{2n}}{2n}$ and $\sum_{n=1}^\infty \frac{x^{2n-1}}{(2n-1)^2}$.