Cauchy's condensation test

Block bounds. The block $2^k\le n<2^{k+1}$ contains $2^k$ terms, each between $u_{2^{k+1}}$ and $u_{2^k}$:
$$2^ku_{2^{k+1}}\le\sum_{n=2^k}^{2^{k+1}-1}u_n\le2^ku_{2^k}.$$

If the condensed series converges, summing the right inequality over $k$ shows $\sum u_n$ is bounded above by $\sum2^ku_{2^k}<\infty$, so it converges (positive terms).

If the condensed series diverges, the left inequality gives $\sum_n u_n\ge\sum_k2^ku_{2^{k+1}}=\frac12\sum_{k\ge1}2^ku_{2^k}$, which diverges.

Application. For $u_n=\frac1{n^p}$ with $p>0$ (decreasing): $2^ku_{2^k}=2^{k(1-p)}$, a geometric series that converges iff $2^{1-p}<1$, i.e. $p>1$. For $p\le0$ the terms do not tend to $0$.