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PutnamSuperchallengeIntegrals I: Feynman's Trick

Integrals I: Feynman's Trick · Superchallenge problem

Compute ∑n=1∞1n(n+1)(n+2)\displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}. Express your answer as a common fraction.