Gamma Product

Calculus Level pending

Find the real number a a such that

∏ k = 1 ∞ ( a k + Γ ( k + 1 k ) ) \prod _{k=1}^{\infty } \left(\frac{a}{k}+\Gamma \left(\frac{k+1}{k}\right)\right)

converges.


The answer is 0.577216.

Christopher Criscitiello by Christopher Criscitiello , 25, USA

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1 solution

Hint: In order for the product to converge, each multiplicand must be 1 + O ( k − 2 ) 1+O(k^{-2}) . Sp choose a a so that this occurs (it might help to expand Γ ( x + 1 x ) \Gamma(\frac{x+1}{x}) in a Laurent series).

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