HMMT February

Algebra Difficulty 4.9 AIME Find the answer

Find the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n^{2}+2n}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We know that $\frac{1}{n^{2}+2n}=\frac{1}{n(n+2)}=\frac{\frac{1}{n}-\frac{1}{n+2}}{2}$ . So, if we sum this from 1 to $\infty$ , all terms except for $\frac{1}{2}+\frac{\frac{1}{2}}{2}$ will cancel out (a 'telescoping' series). Therefore, the sum will be $\frac{3}{4}$ .

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