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?