Serious Series Problem

**bearej50** · February 9th 2010, 03:52 PM

What is the sum of:

1/1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ... + 1/(1+2+3+...+n)

**Drexel28** · February 9th 2010, 03:59 PM

Originally Posted by bearej50

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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