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Sum of a power series

  1. Apr 20, 2014 #1
    1. The problem statement, all variables and given/known data

    find the sum of the following series:

    [itex] \sum_{n=1}^\infty nx^{n-1} , |x|<1 [/itex]

    2. Relevant equations

    [itex] \frac{a}{1-r} [/itex]


    3. The attempt at a solution

    i know that a function representation for that series is [itex] -\frac{1}{(1-x)^2} [/itex] but how is it possible to find the sum of a series with a variable in it? please help :(
     
    Last edited: Apr 20, 2014
  2. jcsd
  3. Apr 20, 2014 #2

    micromass

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    Do you know what

    [tex]\sum_{n=0}^{+\infty} x^n[/tex]

    is?
     
  4. Apr 20, 2014 #3
    [itex] \frac{1}{1-x} [/itex]
     
  5. Apr 20, 2014 #4

    micromass

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    Now take derivatives.
     
  6. Apr 20, 2014 #5
    i know that the series as a function is [itex] \frac{-1}{(1-x)^2} [/itex] but webassign said that was wrong. they are looking for the sum of the series.
     
  7. Apr 20, 2014 #6

    micromass

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    Yes, it is wrong. Please show your work.
     
  8. Apr 20, 2014 #7
    I wrote it as
    [itex] (1-x)^{-1 }[/itex]
    to take the derivative i multiplied it by the exponent and subtracted one from the exponent.
    [itex] -1(1-x)^{-2} [/itex]
    which is
    [itex] -\frac{1}{(1-x)^2} [/itex]
     
  9. Apr 20, 2014 #8
    oh wait i see it now. i forgot to use the chain rule. it should be
    [itex] \frac{1}{(1-x)^2} [/itex]
     
  10. Apr 20, 2014 #9

    micromass

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    Indeed!
     
  11. Apr 21, 2014 #10
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