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PutnamSuperchallengeIntegrals I: Feynman's Trick

Integrals I: Feynman's Trick · Superchallenge problem

Evaluate ∑k=1∞1k(k+1)(k+2)(k+3)\displaystyle\sum_{k=1}^{\infty} \frac{1}{k(k+1)(k+2)(k+3)}. Express your answer as a common fraction.