Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 19 / b

The term at $k=0$ is finite. For every $k\ge 1$,The comparison P-series $\sum k^{-2}$ converges, so the comparison test for series proves that the given series converges absolutely.

For $u_k=a^{2k+1}/(2k+1)>0$, the consecutive ratio isThus the ratio test proves convergence when $0<a<1$ and divergence when $a>1$. At $a=1$, the series is ${\sum }_{k\ge 0}1/(2k+1)$, which diverges by comparison with $\frac{1}{2}{\sum }_{k\ge 0}1/(k+1)$. ThereforeAll terms are nonnegative, so convergence here is also absolute convergence.