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Thread: Telescoping series

  1. #1
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    Telescoping series

    Can someone check my solution:

    Consider the series $\displaystyle \sum_{k=1}^{\infty}a_k$ where $\displaystyle a_k=
    \frac{1}{k(k+1)}$

    1) Find the partial fractions decomposition of $\displaystyle a_k$

    $\displaystyle a_k=\frac{1}{k}-\frac{1}{k+1}$

    2) Find a simple formula for the partial sum $\displaystyle s_n$, where $\displaystyle s_n=\sum_{k=1}^{n}a_k$

    $\displaystyle s_n=\sum_{k=1}^{n}a_k=\sum_{k=1}^{n}\frac{1}{k}-\frac{1}{k+1}$

    $\displaystyle =\sum_{k=1}^{n}\frac{1}{k}-\frac{1}{k+1}$

    $\displaystyle =1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n}-\frac{1}{n+1}$

    $\displaystyle =1-\frac{1}{n+1}$

    3) Hence find

    $\displaystyle \sum_{k=1}^{\infty}a_k$



    $\displaystyle \sum_{k=1}^{\infty}a_k=\lim_{n\to\infty}\sum_{k=1} ^{n}a_k$

    $\displaystyle =\lim_{n\to\infty}\left(1-\frac{1}{n+1}\right)=1$
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  2. #2
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    Looks fine from overe here.
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