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Math Help - Serious Series Problem

  1. #1
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    Serious Series Problem

    What is the sum of:

    1/1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ... + 1/(1+2+3+...+n)
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bearej50 View Post
    What is the sum of:

    1/1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ... + 1/(1+2+3+...+n)
    \sum_{k=1}^{j}k=\frac{j(j+1)}{2}. And so \sum_{j=1}^{n}\frac{1}{\sum_{k=1}^{j}k}=\sum_{j=1}  ^{n}\frac{2}{j(j+1)}=2\sum_{j=1}^{n}\left\{\frac{1  }{j}-\frac{1}{j+1}\right\}=2\left[1-\frac{1}{n+1}\right].
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