Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 19 / b / Solution

Cancellation of the factorials in the consecutive ratio givesSet $t=1/n$. The derivative of the natural logarithm at $1$ gives $\ln (1+t)/t\to 1$. ThereforeThus the necessary condition from the term test for divergence is satisfied. On its own it does not prove that $\sum u_n$ converges.