as .

Writing and comparing with and , integration from to gives

and dividing by shows that .

For twice differentiable and , integration by parts twice gives

Taking , , and summing over ,

where

So the series converges, and

for the constant .

It remains to show that . The Wallis integrals

satisfy by integration by parts, so

Since and is decreasing in , sandwiching gives , hence

On the other hand, Stirling’s asymptotic form with the constant gives . Comparing the two forces .