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Manipulation of Power Series

  1. Nov 22, 2015 #1
    1. The problem statement, all variables and given/known data
    By considering the power series (good for |x| < 1)

    ##\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +....##

    Describe how to manipulate this series in some way to obtain the result:

    ##\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}##
    2. Relevant equations
    Maclaurin series?

    3. The attempt at a solution
    I was trying to somehow relate n to ##\frac{x}{1-x}## and then multiply the first series (with its index shifted) by that result. Problem is I couldn't find any relation. I basically don't even know where to start. I would appreciate a hint.
  2. jcsd
  3. Nov 22, 2015 #2


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    Have you tried taking the first derivative of the first series?
  4. Nov 22, 2015 #3
    No. I'll try that.

    I guess my problem was understanding what they meant by "manipulating". Wouldn't taking the derivative of a series give me a whole new series? I don't see how that's "manipulating".

  5. Nov 22, 2015 #4


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    When you compare the compact forms of the function, you can't see any equivalence between them. They are indeed an entirely different functions. I think this problem just asks you to find a way to get the ##x/(1-x)^2## from ##1/(1-x)## by looking at their respective power series.
  6. Nov 22, 2015 #5


    Staff: Mentor

    By "manipulating" the series, they mean applying some operation to it. Taking the derivative term-by-term definitely counts as manipulating the series.
  7. Nov 22, 2015 #6
    Alright, I took the first derivative and multiplied by x. Thanks guys.
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