AP Calculus BC

Convergence vs Divergence

A series converges when its partial sums approach a finite limit, and that limit is the sum. It diverges in every other case, which covers partial sums that grow without bound and partial sums that oscillate forever without settling on any one value.

Convergence

Use when: The partial sums close in on a single finite number, so the infinite sum has a value you can name, approximate, or bound.

Divergence

Use when: The partial sums run off without bound or keep swinging between values, so there is no sum to report.

Side by side

	Convergence	Divergence
Partial sums $S_N$	Approach a finite limit $S$	Grow without bound or never settle
The infinite sum	Exists and equals $S$	Does not exist
What the terms do	Must satisfy $\lim_{n \to \infty} a_n = 0$	May or may not approach $0$ ; $\sum \frac{1}{n}$ diverges with terms approaching $0$
Standard example	$\sum \frac{1}{n^2}$	$\sum \frac{1}{n}$ and $\sum (-1)^n$
Common trap	Treating shrinking terms as proof	Assuming it always means growing to infinity

Both words describe the partial sums, not the terms. Build $S_N = a_1 + a_2 + \cdots + a_N$ and ask what the sequence $S_N$ does as $N$ grows. If it approaches a finite $S$ , the series converges to $S$ . Everything else is divergence.

Divergence comes in two forms worth separating. The harmonic series grows without bound, its partial sums creeping up like $\ln N$ , slowly but forever. The series $\sum (-1)^n$ does something different: its partial sums bounce between two values and never approach anything, so the terms stay bounded and the series still diverges.

On the exam, name the test

A verdict without a named test earns nothing. Write which test you used, state the condition it needs, and confirm that condition holds. Saying the terms do not approach zero is the nth term test, and it should be called that.

Frequently asked questions

Can a series diverge even though its terms go to zero?

Yes, and the harmonic series is the standard case. Its terms shrink to zero while its partial sums grow without bound, which is why terms approaching zero is necessary for convergence but never sufficient.

Does divergence always mean the sum runs to infinity?

No. Oscillation counts as divergence too. The partial sums of $\sum (-1)^n$ alternate between two values and never approach a limit, so the series diverges even though nothing grows.

Do I need the actual sum to say a series converges?

No. Convergence tests decide the question without ever producing the value. The sums you can actually name are the geometric ones, the telescoping ones, and any series you recognise as a known Maclaurin series evaluated at a particular number.

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