Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 2 / b

The implication is true, with absolute convergence. Since $n^{3/2}{a}_n\to 0$, there is $N$ such that $|a_n|\le n^{-3/2}$ for $n\ge N$. The p-series $\sum n^{-3/2}$ converges because its exponent exceeds one. The comparison test proves $\sum |a_n|<\infty$, and absolute convergence implies convergence of $\sum a_n$.

The implication is false. SetThen $n{a}_n=1/\log (n+1)\to 0$. For $n\ge 2$, $\log (n+1)\le \log (2n)\le 2\log n$, so $a_n\ge 1/(2n\log n)$. The integral test givesThus the comparison test shows that $\sum a_n$ diverges.