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PutnamSuperchallengeIntegrals I: Feynman's Trick

Integrals I: Feynman's Trick · Superchallenge problem

Evaluate ∑n=0∞(2nn)15n\displaystyle\sum_{n=0}^{\infty}\binom{2n}{n}\frac{1}{5^n}. Express your answer in simplest radical form.