Thread: series problem

I need the answer of these series with respect to x. can anyone solve it, please?

It reminds me of the series .

- Hollywood

Let -- notice your function is . Now find and try to recognize this as a geometric series. Then integrate to find your original series. Since this is obviously a text book problem, the answer is "nice".

Yeah, what johng suggests is the way to go. I don't know what I was thinking.

- Hollywood

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?

I think that if I replace n+1 with n+1/2+1/2, and by using the method of johng, I can solve this,too. Thank you. your idea help me to solve my problem.