In AP Calculus BC, a sequence is an ordered list of numbers (terms) indexed by n, like a₁, a₂, a₃, ... A sequence converges if its terms approach a single real number as n goes to infinity, and that idea of a sequence limit is the foundation for defining whether an infinite series converges.
A sequence is an ordered list of numbers. Each number in the list is a term, and we label them with a position index n, so you'll see notation like aₙ or {aₙ}. A sequence can be finite (it stops) or infinite (it keeps going forever), and on the BC exam you almost always care about infinite sequences.
Here's the move that makes sequences matter in calculus. You ask what happens to aₙ as n goes to infinity. If the terms settle in on one real number L, the sequence converges to L. If they don't (they blow up, or they bounce around forever, like alternating between -10 and 10), the sequence diverges. That's it. A sequence is really just a function whose domain is the positive integers, so all your limit intuition from earlier in the course carries over directly.
Sequences live in Unit 10: Infinite Sequences and Series (BC only), starting with Topic 10.1, and they directly support learning objective 10.1.A (determine whether a series converges or diverges). The CED's whole definition of series convergence is built on sequences. Per LIM-7.A.1, the nth partial sum Sₙ is the sum of the first n terms of a series. Per LIM-7.A.2, an infinite series converges to a real number S if and only if the limit of its sequence of partial sums exists and equals S. Read that again. A series converging is defined as a sequence converging. So if you're shaky on sequences, every convergence test in Unit 10 (and the whole Taylor series machinery in Units 10.11-10.15) is built on sand.
Sequence of Partial Sums (Unit 10)
This is the bridge between sequences and series. Take a series, chop it off after 1 term, 2 terms, 3 terms, and so on. Those running totals S₁, S₂, S₃, ... form a sequence, and the series converges exactly when that sequence does. Every series question is secretly a sequence question.
Convergence and Divergence (Unit 10)
Convergence for a sequence means the terms aₙ approach one number. Convergence for a series means the partial sums approach one number. Keeping those two layers straight is half the battle in Unit 10, because a sequence can converge (like aₙ = 1/n going to 0) while the series built from it diverges (the harmonic series).
Limits at Infinity (Unit 1)
Evaluating lim aₙ as n → ∞ uses the exact same toolkit as lim f(x) as x → ∞ from Unit 1. Dominant terms, ratios of polynomials, exponentials beating powers. A sequence is just a function sampled at the integers, so nothing here is new, it's just wearing an n instead of an x.
Monotonic Sequence (Unit 10)
A monotonic sequence always moves one direction (always increasing or always decreasing). Monotonic plus bounded guarantees convergence, which is a quick structural argument for why certain sequences and series settle down.
Sequences show up on BC multiple choice as direct limit evaluations. A typical stem hands you a few sequences and asks which one converges, or gives you a sequence like one alternating between -10 and 10 and asks for its limit (answer: the limit doesn't exist, so it diverges). The bigger payoff is indirect. Series FRQs, which appear on nearly every BC exam, lean on the partial-sum definition from LIM-7.A.2 and on the nth term test, both of which are sequence-limit arguments. You need to do two things: compute lim aₙ as n → ∞ cleanly, and state precisely whether a sequence converges or diverges, with the right justification language (the limit "exists and equals L" versus "does not exist").
A sequence is a list of numbers; a series is the sum of a sequence's terms. The sequence 1/2, 1/4, 1/8, ... converges to 0, while the series 1/2 + 1/4 + 1/8 + ... converges to 1. Different objects, different answers. The classic trap is the harmonic case, where the sequence 1/n converges to 0 but the series Σ 1/n diverges. Terms going to 0 is necessary for a series to converge, but never sufficient.
A sequence is an ordered list of numbers indexed by n, and you can treat it as a function whose domain is the positive integers.
A sequence converges if lim aₙ as n → ∞ exists and equals a single real number; otherwise it diverges.
A sequence that oscillates between two values forever, like -10 and 10, has no limit and therefore diverges.
By LIM-7.A.2, a series converges to S if and only if its sequence of partial sums converges to S, so series convergence is defined through sequence convergence.
The terms of a sequence going to 0 does not mean the series of those terms converges; the harmonic series is the standard counterexample.
Sequences and series are tested on AP Calculus BC only, in Unit 10.
A sequence is an ordered list of numbers, where each number is called a term and gets an index n (written aₙ). In Unit 10 you mostly study infinite sequences and ask whether their terms approach a limit as n goes to infinity.
A sequence is a list of numbers; a series adds those numbers up. For example, the sequence 1/2, 1/4, 1/8, ... converges to 0, but the series 1/2 + 1/4 + 1/8 + ... converges to 1. The AP exam expects you to keep these two objects separate.
No. Terms going to 0 is necessary but not sufficient for series convergence. The harmonic series Σ 1/n is the famous counterexample: the sequence 1/n converges to 0, but the sum grows without bound.
No. Since the terms bounce between two values forever and never settle on one number, the limit as n approaches infinity does not exist, so the sequence diverges. This is a common multiple-choice setup.
No. Sequences and series are Unit 10 content, which is BC only. If you're taking AB, you won't see this material; if you're taking BC, expect it on both multiple choice and a free-response question most years.