Schlömilch's generalization of the condensation test

Remarks

The proof cuts the series into blocks $u_k \le n < u_{k+1}$ of $\Delta u_k$ terms each. Since $(a_n)$ decreases, the block sum lies between $\Delta u_k \, a_{u_{k+1}}$ and $\Delta u_k \, a_{u_k}$. The upper bound shows that convergence of the condensed series gives convergence of $\sum a_n$. The lower bound and $\Delta u_{k+1} \le M \Delta u_k$ show that $\Delta u_{k+1} a_{u_{k+1}}$ is at most $M$ times the block sum, which gives the converse. With $u_k = 2^k$ this is the Cauchy condensation test. With $u_k = k^2$ for $k \ge 1$ the condensed series is $\sum (2k+1) a_{k^2}$, and the condition holds since $(2k+3)/(2k+1) \le 3$. For $a_n = 1/n^p$ with $p > 0$ this gives $\sum (2k+1)/k^{2p}$, which converges exactly when $2p - 1 > 1$, that is $p > 1$.

Going deeper

For a student

The Cauchy condensation test uses the gaps $2^k$. Schlömilch's version lets you choose gaps adapted to the series, such as the squares, which helps when $a_{k^2}$ has a simpler form than $a_{2^k}$. The hypothesis that consecutive gaps have bounded ratio is what makes the lower bound work. The proof is the same comparison of block sums as in the Cauchy case.