Binomial Limit

Q: How can I compute the limit of as through integer values?

A: Numerically, it appears that the limit is one.

Analytically, we can use FunctionExpand to replace the binomial by gamma functions.

Next, we use the reflection formula, , here implemented via pattern matching.

This expression simplifies, noting that .

Since we are interested in the asymptotic limit, , we use Series (effectively using Stirling’s formula).

Noting that , followed by computing the limit as , gives us the desired result.