The Integral Test

Calculus 2 · Axiom Academy

When a series and an integral share the same fate: a bridge between a discrete sum and a continuous area. Take a series where f is positive and decreasing. Each term f(n) is the height of a rectangle of width 1 , so the series is a staircase of rectangles. The improper integral is the shaded area under the smooth curve y=f(x) . Watch how the rectangle heights are read straight off the curve at the integers. Because f decreases, the same rectangles can be anchored two ways. The animation draws both families off the one curve, so you can see each rectangle's height come straight from f at an integer. Rectangles sit above the curve — an overestimate of the integral. Rectangles sit below the curve — an underestimate of the integral. Trapping the area between the two staircases gives the inequality the test runs on: The squeeze from Step 2 forces it: the series partial sums and the integral differ by less than f(1) , so neither can run off to infinity without dragging the other along. The animation races a convergent case against a divergent one — both deciding the curve and its rectangles at once. The headline application is the p‑series. Here f(x) = 1/x^p is positive and decreasing for , so the Integral Test applies and we just have to evaluate one integral: The animation lines up three p‑values and grows each series' partial sums as bars: keeps climbing without bound, while flattens onto a finite ceiling — exactly what the integrals predicted.

This is the written version of the interactive lesson above. See the full Calculus 2 course.