MathDB

10

Part of 2023 IMC

Problems(1)

The denominator is big

Source: 2023 IMC P10

8/3/2023
For every positive integer nn, let f(n)f(n), g(n)g(n) be the minimal positive integers such that 1+11!+12!+⋯+1n!=f(n)g(n).1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}. Determine whether there exists a positive integer nn for which g(n)>n0.999ng(n)>n^{0.999n}.
number theoryasymptotic behaviourIMCcollege contestsIMC 2023