AP Calculus BC glossary

Convergent sequence

Also called: Sequence convergence

A sequence is convergent when the limit of its terms, as the index runs to infinity, exists and is finite. Every convergent sequence is bounded, but bounded sequences need not converge. A convergent sequence of terms does not make the series built from them converge.

Deciding convergence is an ordinary limit at infinity, and the techniques for evaluating one belong to the limit of a sequence. What convergence itself adds is that only the tail matters: changing or dropping finitely many terms changes neither the verdict nor the limit. The standard case is the geometric sequence rnr^n, which converges for −1<r≤1-1 < r \le 1 and diverges for every other value of rr.

Convergence of the sequence ana_n and convergence of the series ∑an\sum a_n are separate questions with separate answers. The sequence 1n\frac{1}{n} converges to 00 while the series built from it diverges. Every convergent sequence is bounded, but bounded sequences such as (−1)n(-1)^n need not converge.

The mistake

Concluding a series converges because its terms converge to zero. A series converges when the sequence of partial sums SNS_N converges, and that is a different sequence. The terms of the harmonic series converge to 00 while its partial sums grow past every bound. Checking the terms is still the right first move, but it can only ever prove divergence.

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