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Math Help - alternating series

  1. #1
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    alternating series

    Hi,
    Can somebody please tell me how to find the sum of the following alternating series?


    \sum_{n=0}^{\infty} (-1)^n exp(-5*n)/(2*n+1)

    Thanks,
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  2. #2
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    Re: alternating series

    \sum_{n=0}^{\infty} (-1)^n \frac{e^{-5n}}{2n+1}
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  3. #3
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    Re: alternating series

    Quote Originally Posted by rvoy View Post
    Can somebody please tell me how to find the sum of the following alternating series?
    \sum_{n=0}^{\infty} (-1)^n exp(-5*n)/(2*n+1)

    Is this your question \sum\limits_{n =1}^\infty  {\frac{{{{\left( { - 1} \right)}^n}}}{{{e^{5n}}\left( {2n + 1} \right)}}}~?

    If so, can you show that \left( {\frac{1}{{{e^{5n}}\left( {2n + 1} \right)}}} \right) is a monotone decreasing null sequence?
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  4. #4
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    Re: alternating series

    Hi Plato,
    Yes this is the series,

    I know this series is absolutely convergent as it is bounded above by a geometric series. But, my question is how to find the sum
    of the given alternating series.
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  5. #5
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    Re: alternating series

    If you take the integral of the geometric series

    \sum_{n=0}^\infty x^{2n}\,dx

    term-by-term, you get

    \sum_{n=0}^\infty \frac{x^{2n+1}}{2n+1}

    That gives you the 2n+1 in the denominator. I think that by setting x to a strategic value and multiplying by constants inside and outside of the sum, you'll be able to come up with the answer.

    - Hollywood
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