AP Calculus BC glossary

Divergence

A series diverges when its partial sums fail to approach a finite limit. That can mean the sums grow without bound, as in the harmonic series, or that they never settle at all, as when the terms do not shrink to zero.

The nth term test is the only test that can prove divergence on its own from the terms alone. If the terms do not approach zero the series diverges; if they do, the test is silent.

The mistake

Concluding convergence because the terms go to zero. The harmonic series has terms tending to zero and still diverges, which is why the nth term test never proves convergence.

Appears in: Unit 10: Infinite Sequences and Series (BC)