# The exponential grows faster than polynomials

Earlier this week, I was trying to correctly write up why the gamma function converges for and I realised that I did not know how to show that the exponential function grows faster than any polynomial function. It’s only just a simple application of the Taylor expansion of the exponential but I’ll write it up since I’m sure a lot of people never saw the proof.

First, what I mean by that is that for any there exists an such that

for any To show this, we will show that

This suffices because then for large enough this ratio will be greater than and therefore we will have In fact, we could only show that this limit is greater than 1.

First, note that

as This is often written Thus, for any when is large enough we have

Therefore, we only need to show that

Since this limit goes to infinity iff goes, we will drop the in the denomiator.

Expanding in its Taylor series, we get

which tends to as

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