Let $\displaystyle f(x)=\sum^\infty_{i=0}(-1)^n{1\over (2n+1)3^n}x^{2n+1}$ -- notice your function is $\displaystyle {1\over x}f(x)$. Now find $\displaystyle f^\prime(x)$ and try to recognize this as a geometric series. Then integrate $\displaystyle {1\over x}f^\prime(x)$ to find your original series. Since this is obviously a text book problem, the answer is "nice".
Thank you very much. It works for this series. I found out this problem by simplifying a bigger problem in my master thesis in finance. It is not from a textbook. But, unfortunately I made a mistake while I was simplifying the problem. There is an additional term in numerator. It means that problem is:
(n+1)/(2n+1)*(-x^2/3)^n
It cannot be recognized as a geometric series by the same method. Is there any solution for my problem?