Integral Test for Convergence

A series is just a sum of areas of unit-width rectangles, one for each term. If the function generating those terms is positive and decreasing, you can line the rectangles up against the curve y = f(x) and compare their total area to the area under the curve itself. Since the curve runs forever, asking whether its area is finite is the same question as asking whether the integral from some starting point to infinity converges. That improper integral becomes a stand-in for the whole infinite sum.

The mechanism is a squeeze. Draw rectangles starting just after each integer, and they sit below the curve, so the sum from n = 2 to infinity is less than the integral from 1 to infinity. Draw them starting at each integer instead, and they sit above the curve, so the sum is greater than the integral from 1 to infinity. Either way, the series and the integral rise or stay finite together: if the integral converges, the series is squeezed into a finite value too, and if the integral diverges, the series is forced to diverge as well. This only works when f is positive, continuous, and decreasing, because that is what keeps the rectangles pinned to one side of the curve.

This test matters most when terms do not fit a geometric or p-series pattern directly but come from a function you can actually integrate, like 1/(n ln n). It sits alongside the p-series test and comparison tests as another way to pin down convergence, and the substitution tricks you use here, like u = ln(x), are the same ones that show up again when series involve logs or exponentials.