look at this series:
$\sum\limits_{n = 1}^\infty {\dfrac{{{{( - 1)}^{n - 1}}}} {{2n - 1}}{x^{2n}}}$
by Cauchy-Hadamard formula,the above series convergence region is $(-1,1)$. at the end points, it is a alternating series. so convergence.so its convergence region is $[-1,1]$.
my question is: how to find a function which Taylor expansion equal the above series?
thanks very much