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Math Help - Sum of mixed series

  1. #1
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    Sum of mixed series

    I have the series 1r, 2r^2,3r^3...(or in general nr^n)
    what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

    Thanks for your help
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  2. #2
    Super Member girdav's Avatar
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    Re: Sum of mixed series

    Let f(r) :=\sum_{n=1}^{+\infty}nr^n. You can integrate it term-by-term, for example from 0 to x, with x<1. Why?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Sum of mixed series

    Quote Originally Posted by Schdero View Post
    I have the series 1r, 2r^2,3r^3...(or in general nr^n)
    what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

    Thanks for your help
    Is...

    \sum_{n=1}^{\infty} n\ r^{n} = r\ \sum_{n=1}^{\infty} n\ r^{n-1} = r\ \frac{d}{d r}\ \sum_{n=1}^{\infty} r^{n} (1)

    ... and, because for |r|<1...

    \sum_{n=1}^{\infty} r^{n} = \frac{1}{1-r} - 1 (2)

    ... we have for |r|<1 ...

    \sum_{n=1}^{\infty} n\ r^{n} = \frac{r}{(1-r)^{2}} (3)

    Kind regards

    \chi \sigma
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  4. #4
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    Re: Sum of mixed series

    Thank you for the quick reply! What if, e.g., im looking for the sum up to only n=8 though?
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  5. #5
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    Re: Sum of mixed series

    Quote Originally Posted by Schdero View Post
    I have the series 1r, 2r^2,3r^3...(or in general nr^n)
    what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

    Thanks for your help
    You can get to a standard geometric series as follows....

    S_n=r+2r^2+3r^3+.....+nr^n

    rS_n=r^2+2r^3+3r^4+....+nr^{n+1}

    S_n-rS_n=r+r^2+r^3+....r^n-nr^{n+1}

    S_n(1-r)=\left(r+r^2+r^3+...+r^n\right)-nr^{n+1}

    The geometric series in brackets can now be evaluated
    and hence a closed form for the sum can be found.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Sum of mixed series

    Another way: denoting S=r+2r^2+3r^3+\ldots\quad , we have rS=r^2+2r^3+3r^3+\ldots . This implies

    (1-r)S=r+r^2+r^3+\ldots=\dfrac{1}{1-r}-1=\dfrac{r}{1-r}\quad (|r|<1)

    So, S=\dfrac{r}{(1-r)^2}


    Edited: Sorry, I didn't see Archie Meade's post
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  7. #7
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    Re: Sum of mixed series

    Thank you very much to all of you!!
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