Is there a closed form for the following infinite product?

∞∏n=12n√Γ(2n+12)Γ(2n)

**Answer**

The beautiful idea of Raymond Manzoni can actually be made rigorous. Consider a **finite** product ∏Ln=1 and take its logarithm. After using duplication formula for the gamma function and telescoping, it simplifies to the following:

L∑n=112nlnΓ(2n+12)Γ(2n)=(1−2−L)ln(2√π)−2Lln2+2⋅12L+1lnΓ(2L+1).

This is an exact relation, valid for any L. Now it suffices to use Stirling,

1NlnΓ(N)=lnN−1+O(lnNN)asN→∞

to get

∞∑n=112nlnΓ(2n+12)Γ(2n)=ln(2√π)+2(ln2−1)=ln8√πe2.

So the answer is indeed 8√πe2.

**Attribution***Source : Link , Question Author : Laila Podlesny , Answer Author : Start wearing purple*