Cauchy Condensation Test

Cauchy Condensation Test: Let $(a_n)$ be a nonincreasing sequence of nonnegative real numbers. Then $\sum_{n=1}^\infty a_n \text{ converges } \Longleftrightarrow \sum_{k=0}^\infty 2^k a_{2^k} \text{ converges}.$

This test is especially useful for series like $\sum 1/n^p$ and $\sum 1/(n(\log n)^p)$.

Proof sketch (optional): Group terms in dyadic blocks $[2^k,2^{k+1}-1]$ and use monotonicity to bound each block between $2^k a_{2^{k+1}}$ and $2^k a_{2^k}$.