Asymptotic Expansion of a Sum

Q: Let be an arbitrary irrational number. How can I find an asymptotic approximation to

as approaches 1 from below?

A: The sum can be expressed as a Gaussian hypergeometric function.

There are at least three advantages of this representation:

1. Mathematica includes arbitrary-precision algorithms for computing this function. For example, here is the sum with and .

2. Asymptotic approximations can be computed via series expansion. Introducing , the following expansion is useful when .

Here is the asymptotic approximation with and .

3. The asymptotic series expansion in closed form can be obtained using functions.wolfram.com/07.23.06.0012.01,

Here is the result.

This can be used to compute the number of asymptotic terms required. For example, terms are required to get a result accurate to at least 10 decimal places with and .