p-Series

A p-series looks like 1/1^p + 1/2^p + 1/3^p + ... for some fixed power p. The question is always the same: does making infinitely many of these tiny fractions still add up to something finite? The answer hinges entirely on p. If p > 1, the terms shrink fast enough that the total settles on a finite number. If p ≤ 1, the terms don't shrink fast enough, and the sum grows without bound, even though each term individually goes to zero.

The integral test explains why p = 1 is the exact cutoff. Compare the series to the integral of x^−p from 1 to infinity, since the series and the integral rise or fall together (both are measuring the area under a decreasing curve made of the same shrinking pieces). That integral evaluates to a finite number only when p > 1 — the antiderivative x^1-p/(1-p) goes to 0 as x goes to infinity exactly when the exponent 1-p is negative, meaning p > 1. At p = 1 the integral becomes ln(x), which grows forever, so the harmonic series sum of 1/n diverges even though 1/n → 0.

P-series are the benchmark for the whole unit. Once you know their fate, you can judge far messier series by comparison: if a complicated series behaves like 1/n² for large n, it converges; if it behaves like 1/n, it diverges. The comparison test and limit comparison test, which come right after this topic, lean on p-series as the known reference point.