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Math Help - sum of infinite series (clarification)

  1. #1
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    sum of infinite series (clarification)

    I have a series from n=1 to infinity of

    n/(4^n).

    Can this convert into a geometric series? and so how?
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  2. #2
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    Quote Originally Posted by guyonfire89 View Post
    I have a series from n=1 to infinity of
    n/(4^n).
    Can this convert into a geometric series? and so how?
    Yes it can by differentiation.
    Start with  \displaystyle f(x) = \sum\limits_{n = 0}^\infty  {x^n }  = \frac{1}{{1 - x}},~|x|<1.

    Differentiate to show  \displaystyle f'(x) = \sum\limits_{n = 1}^\infty  {nx^n }  = \frac{x}{{(1 - x)^2}}

    Then let x=\frac{1}{4}
    Last edited by Plato; September 20th 2010 at 02:59 PM.
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    Can someone explain how x=1/4?
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    Quote Originally Posted by guyonfire89 View Post
    Can someone explain how x=1/4?
    You let x = \frac{1}{4} and the LHS, that's the sum \sum\limits_{n = 1}^\infty {nx^n } becomes \sum\limits_{n = 1}^\infty {\frac{n}{4^n} } which is your original sum.
    Do the same, i.e. let x = \frac{1}{4} for the RHS, that's \dfrac{x}{{(1 - x)^2}} and you will find what it evaluates to.
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