AP Calculus BC

Sequence vs Series

A sequence is an ordered list of numbers; a series is the sum of that list. A sequence converges when its terms approach a limit, while a series converges when its partial sums approach a limit, which is a much stronger requirement.

Sequence

Use when: You are asked what the terms approach, or whether the terms themselves settle down.

Series

Use when: You are asked whether a sum converges, or for the value of an infinite sum.

Side by side

	Sequence	Series
Object	A list	A sum
Notation	$a_n$	$\sum a_n$
Converges when	The terms approach a limit	The partial sums approach a limit
Harmonic case	$\frac{1}{n} \to 0$ , converges	$\sum \frac{1}{n}$ diverges

The harmonic case is the whole reason the distinction matters. The terms shrink to zero, so the sequence converges, yet the partial sums grow without bound and the series diverges. Terms going to zero is necessary for a series to converge but nowhere near sufficient.

That asymmetry is exactly what the nth term test encodes: it can prove divergence when the terms do not approach zero, and it can never prove convergence.

Frequently asked questions

If a sequence diverges, does its series diverge?

Yes. If the terms do not approach zero the series cannot converge, which is the nth term test.

In the CED: Unit 10: Infinite Sequences and Series (BC)