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Math Help - series problem

  1. #1
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    series problem

    I need the answer of these series with respect to x. can anyone solve it, please?
    series problem-series.png
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  2. #2
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    Re: series problem

    It reminds me of the series \sin{x} = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1}.

    - Hollywood
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  3. #3
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    Re: series problem

    Let f(x)=\sum^\infty_{i=0}(-1)^n{1\over (2n+1)3^n}x^{2n+1} -- notice your function is {1\over x}f(x). Now find f^\prime(x) and try to recognize this as a geometric series. Then integrate {1\over x}f^\prime(x) to find your original series. Since this is obviously a text book problem, the answer is "nice".
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  4. #4
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    Re: series problem

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

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  5. #5
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    Re: series problem

    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?
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  6. #6
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    Re: series 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.
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