AP Calculus BC glossary

Series

Also called: Infinite series

A series is the sum of the terms of a sequence, often infinitely many. It converges when the sequence of its partial sums approaches a finite limit, and that limit is defined to be the sum of the series.

∑n=1∞an=lim⁡N→∞∑n=1Nan\sum_{n=1}^{\infty} a_n = \lim_{N \to \infty} \sum_{n=1}^{N} a_n

Adding infinitely many numbers is not defined directly. The definition routes through partial sums, which are finite additions, and then takes a limit, so every question about a series is really a question about a limit.

Deciding convergence is what the battery of tests is for, and picking the right one is most of the skill. Geometric and p-series can be settled by inspection.

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