Finding the Sum of a Power Series

  1. I'm trying to find the sum of this:

    [tex]
    \[
    \sum\limits_{n = 0}^\infty {( - 1)^n nx^n }
    \]
    [/tex]

    This is what I have so far:

    [tex]
    \[
    \begin{array}{l}
    \frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\
    \frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } = \sum\limits_{n = 1}^\infty {nx^{n - 1} } \\
    \frac{x}{{(1 - x)^2 }} = \sum\limits_{n = 1}^\infty {nx^n } \\
    \end{array}
    \]
    [/tex]

    So how do I get the (-1)^n part in there? Any suggestions would be really helpful. Thanks.
     
  2. jcsd
  3. [tex] \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^{n}x^{n} [/tex]
     
    Last edited: Nov 27, 2006
  4. Oh, I see.. Where exactly does that come from?
     
  5. [tex] \frac{1}{1+x} = \frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^{n} = (-1)^{n}x^{n} [/tex]
     
  6. Ah! Of course. Okay. Thanks.
     
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