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Math Help - Finding the sum of a series

  1. #1
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    Finding the sum of a series

    Hi. I need help showing the work for the sum of the series (N)/(2^N) from N=1 to infinity. The answer is two, but I'm not quite sure how to go about it.
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  2. #2
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    \sum\limits_{k = 0}^\infty  {x^k }  = \frac{1}{{1 - x}},\;\left| x \right| < 1
    \sum\limits_{k = 1}^\infty  {kx^{k - 1} }  = \frac{1}{{^{\left( {1 - x} \right)^2 } }},\; \Rightarrow \sum\limits_{k = 1}^\infty  {kx^k }  = \frac{x}{{^{\left( {1 - x} \right)^2 } }}

    Let x = \frac{1}{2}.
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  3. #3
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    Hello, SnowWhite!

    Another approach . . .


    Find: . \sum^{\infty}_{n=1}\frac{n}{2^n}

    We have: . . . . . S \;\;=\;\;\frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \frac{4}{2^4} + \frac{5}{2^5} + \cdots

    Multiply by \frac{1}{2}\!:\;\;\frac{1}{2}S \;\;=\;\;\qquad\frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \frac{4}{2^5} + \cdots

    Subtract: . . . . \frac{1}{2}S \;\;= \;\;\underbrace{\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \frac{1}{2^5} + \cdots}_{\text{geometric series, sum 1}}<br />

    Therefore: . \frac{1}{2}S\:=\:1\quad\Rightarrow\quad\boxed{S \:=\:2}

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  4. #4
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    Thank you!

    Thank you both so much!
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