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The denominator is big

Source: 2023 IMC P10

August 3, 2023

number theoryasymptotic behaviourIMCcollege contestsIMC 2023

Problem Statement

For every positive integer

n

, let

f(n)

g(n)

be the minimal positive integers such that

1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.

Determine whether there exists a positive integer

n

for which

g(n)>n^{0.999n}