p-series

A p-series tests how the rate at which terms shrink decides whether their sum is finite. You add up reciprocals of powers: the bigger the exponent p, the faster the terms fall, and at some sharp threshold the sum tips from infinite to finite.

The p-series is sum_{n=1}^infinity 1/n^p, where p is a fixed real number. The clean rule is: it converges if and only if p > 1, and diverges if p <= 1. The boundary case p = 1 is the harmonic series, which diverges; p = 2 gives the convergent sum 1 + 1/4 + 1/9 + ... = pi^2/6. The threshold p = 1 is genuinely a knife-edge — convergence fails for p = 1 but holds for every p > 1, no matter how close to 1.

The cleanest proof is the integral test: 1/x^p is positive and decreasing for x >= 1, and the improper integral of 1/x^p from 1 to infinity converges exactly when p > 1. Because the p-series spans the full range of decay rates, it is the benchmark family used by the comparison and limit comparison tests to classify many other series.