integral test

The integral test exploits a simple picture: a series of decreasing positive terms is like a staircase of rectangles sitting under (and over) the graph of a decreasing function. The total area of the rectangles — the series — is trapped between two copies of the area under the curve, so the series and the improper integral converge or diverge together.

Precisely, let f be a positive, decreasing, continuous function on [1, infinity) with f(n) = a_n. Then sum_{n=1}^infinity a_n converges if and only if the improper integral of f from 1 to infinity is finite. The rectangle estimate even gives error bounds: the remainder after N terms lies between the integral of f from N+1 to infinity and the integral from N to infinity, which is handy for numerical estimation.

Two honest conditions must hold: the terms must be eventually positive and eventually decreasing — the test says nothing if f wiggles. It is the cleanest route to the p-series rule, since the integral of 1/x^p over [1, infinity) is finite exactly when p > 1, and it likewise settles borderline series such as sum 1/(n (ln n)^p).