Document Type

Poster

Abstract

A standard calculus problem is to compare the growth rates of n!, nⁿ, and aⁿ, where a > 1 is a parameter. Here, we consider the growth rate of f(x) = (ax)! as x → ∞, generalizing to the gamma function Γ(ax + 1) to allow for non-integer values of ax. We compare the growth rate of f(x) to that of seven other exponential functions, producing a hierarchy of growth rates.

We also further investigate the behavior of the function g(x) = xˣ / (ax+1), giving two different proofs that limx→∞ g(x) = 0. We also consider the locations and values of the local maximum of g(x) as a → 1+ and show an intriguing connection with the Euler-Mascheroni constant.

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Publication Date

2018

Keywords

Calculus, Growth rates, Euler-Mascheroni constant

Disciplines

Mathematics

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