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Homework Help: Power series

  1. May 7, 2006 #1
    Find the sum of the series:
    [tex]\sum\limits_{n = 1}^\infty {nx^n } [/tex] if [tex]
    \left| x \right| < 1

    [/tex]


    I thought maybe with the geometric form, but im not sure.
     
    Last edited: May 7, 2006
  2. jcsd
  3. May 7, 2006 #2
    Is it asking for a number, or just if the series converges?
     
  4. May 7, 2006 #3
    I think for a general solution. It should converge.
     
  5. May 7, 2006 #4
    Is it asking you to find a power series representation??
     
  6. May 7, 2006 #5
    It already is a power series... It's asking for an expression for the sum.
     
  7. May 7, 2006 #6
    You might start by writing out partial sums and see if that gets you anywhere.....
     
  8. May 7, 2006 #7

    shmoe

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    How does this differ from your usual geometric series?
     
  9. May 7, 2006 #8

    Curious3141

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    Hint : Call the original series S. Write out the first five or so terms in the series. Divide the series by x to get a new series (S/x). Now take the difference between this new series and the original series (S/x - S), term by term and see what you end up with.

    The other way to do it is to differentiate a geometric series, but that's more complicated and unnecessary.
     
    Last edited: May 7, 2006
  10. May 8, 2006 #9

    Pyrrhus

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    Last edited by a moderator: Apr 22, 2017
  11. May 8, 2006 #10

    Curious3141

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    Comparing series like these to derivatives of geometric series is a nice and interesting approach (I used to do this), but in most cases I've found that simply dividing or multiplying by x is an easier approach. :smile:
     
    Last edited by a moderator: Apr 22, 2017
  12. May 8, 2006 #11

    shmoe

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    May as well have a third approach:

    [tex]\sum_{n=1}^{\infty}nx^n=\sum_{n=1}^{\infty}\sum_{i=1}^{n}x^n[/tex]

    Change the order of summation (absolutely convergent series) then apply geometric series a couple of times. This is maybe the most complicated of the three, practice in rearranging summations never hurt.
     
    Last edited: May 8, 2006
  13. May 8, 2006 #12
    thanks for all the hints.

    The method I had to use is the derivative of the geometric series (similar to the one used for the maclaurin problem) using

    [tex]

    \left( {\frac{1}{{1 - x}}} \right)^\prime = \sum\limits_{n = 0}^\infty {nx^{n - 1} }


    [/tex]
     
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