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Math Help - Series

  1. #1
    Senior Member slevvio's Avatar
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    Series

    Let  S = 1 + 2x + 3x^{2} + ... + nx^{n-1} . By considering S - xS, show that

     S = \frac{1 - x^{n}}{(1-x)^{2}} - \frac{nx^{n}}{1-x}.

    Hence find the sum of the first n terms of the series:

    1)  \frac{1}{2} + \frac{2}{4} + \frac {3}{8} + \frac{4}{16} + ...,

    2) 1 + 11 + 111 + 1111 + ...


    I can prove that  S = \frac{1 - x^{n}}{(1-x)^{2}} - \frac{nx^{n}}{1-x} but I am not sure how to use the result to work out the sum of the following series. Thanks for any help
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  2. #2
    MHF Contributor

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    Here is a hint on #1.
    \begin{array}{l}<br />
 S = 1 + 2x + 3x^2  +  \cdots  + nx^{n - 1}  \\ <br />
 xS = x + 2x^2  + 3x^3  +  \cdots  + nx^n  \\ <br />
 x = \frac{1}{2} \Rightarrow \frac{1}{2} + \frac{2}{{2^2 }} + \frac{3}{{2^3 }} +  \cdots  + \frac{n}{{2^n }} = ? \\ <br />
 \end{array}
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