Mathematica Journal
Volume 9, Issue 4


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Edited by Paul Abbott

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,

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 .

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