The Divergence Test

Calculus 2 · Axiom Academy

A fast first check: if a series' terms don't shrink to zero, the series must diverge — but watch out, the converse is false. 1. The Divergence Test, Stated The idea is simple. For a sum to settle on a finite value, the pieces you keep adding must eventually become negligible. If instead the terms hover at some level , then every step nudges the running total by about L — and the partial sums march off to infinity. Check the limit of the terms by dividing numerator and denominator by n : The terms head toward , not 0 . By the Divergence Test the series diverges — no further work needed. Watch the terms settle onto the line : 3. The Catch: the Converse Is False This is a one-way test. It can prove divergence, but it can never prove convergence. When the test says nothing — you must reach for another tool (integral test, comparison, ratio, root). The animation walks the valid direction, then shows the reverse direction blocked: 4. The Harmonic Series: the Counterexample So the Divergence Test is inconclusive here. And yet the harmonic series diverges — its partial sums grow without bound, just very slowly. Below, the terms 1/n fall to zero (top track) while the partial sums H_n keep climbing past each gridline (bottom): The whole test in one decision: The test is conclusive — you're done. It might converge or diverge. Use another test to decide. Start here. The Divergence Test is quick — apply it first. If it shows divergence, stop — the series diverges.

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