A telescoping partial-fraction sum

Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n(n+1)(n+2)}$ . (to 4 decimals)

Show hints (2)+

Write $\tfrac{1}{n(n+1)(n+2)} = \tfrac12[\tfrac{1}{n(n+1)} - \tfrac{1}{(n+1)(n+2)}]$ .
The sum telescopes to $\tfrac12\cdot\tfrac{1}{1\cdot2}$ .

Answer

Reveal answer →

0.25 (± 0.0005)

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