Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 9E / b / Solution

Let $S_n={\sum }_{i\le n}{a}_i$ and $T_n={\sum }_{i\le n}|a_i|$. Conditional convergence gives $S_n\to S$ and $T_n\to \infty$. Since $P_n=T_n+S_n$ and $N_n=T_n-S_n$,

The alternating-series test says that $b_n\downarrow 0$ implies convergence of $\sum (-1{)}^n{b}_n$. The hypothesis implies $b_n/b_{n+1}>1$ eventually, so $b_n$ is eventually decreasing. For some $c>0$, eventually $b_n/b_{n+1}\ge 1+c/n$; the divergent product of these factors forces $b_n\to 0$. The test now applies.