International Mathematics Competition
for University Students
2023

IMC2023: Day 2, Problem 10

Problem 10. For every positive integer $\displaystyle n$, let $\displaystyle f(n),g(n)$ be the minimal positive integers such that

$\displaystyle 1+\frac{1}{1!}+\frac{1}{2!}+\ldots+\frac{1}{n!} = \frac{f(n)}{g(n)}.$

Determine whether there exists a positive integer $\displaystyle n$ for which $\displaystyle g(n)>n^{0.999\, n}$.

Fedor Petrov, St. Petersburg State University

  Solution