The Integral Test

Real Analysis · Axiom Academy

When a positive, decreasing f has a_n = f(n) , the series and the integral share one fate — trapped between the same rectangles. 1. A Series Is a Stack of Rectangles Take a function f that is positive, continuous, and decreasing on , and read its values at the whole numbers: a_n = f(n) . Each term becomes a rectangle of width 1 and height a_n , planted on the x -axis right beside the curve y = f(x) . f(x) > 0 for every ( positive ) Here is the whole idea. Because f decreases , the very same rectangles can be placed two ways. Slide them so each sits at its right endpoint and they tuck under the curve; slide them to the left endpoint and they poke over it. The shaded area under the curve never moves — the rectangles just under- and over-shoot it. The tucked-under rectangles give . The poking-over rectangles give . The series is pinned between the integral and the integral plus one term. 3. Reading Off the p -Series Rule Now feed the test the family , giving the p -series . Its fate is the fate of , and that improper integral is easy: it is finite exactly when p > 1 . Watch the tail area as p grows. decays too gently; the tail area is infinite, so the series diverges (e.g. the harmonic series at p=1 , and at ). drops off quickly; the tail area is finite ( ), so the series converges (e.g. at p=2 ). You've seen why a positive, decreasing series and its matching improper integral can never disagree — and read the p -series rule straight off the rectangles.

This is the written version of the interactive lesson above. See the full Real Analysis course.