The sequence of partial sums is the sequence {S_n} where each S_n adds the first n terms of an infinite series; per LIM-7.A.2, the series converges to a sum S if and only if the limit of {S_n} exists and equals S. It's the official AP definition of what a series "adding up" actually means.
Take any infinite series Σa_n and start keeping a running total. S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3, and so on. Each S_n is the nth partial sum (LIM-7.A.1), and the list S_1, S_2, S_3, ... is the sequence of partial sums. Here's the move that makes Unit 10 work. You can't literally add infinitely many numbers, so calculus defines the sum of a series as the limit of this running total. By LIM-7.A.2, the series converges to S if and only if lim(n→∞) S_n = S. If that limit doesn't exist, the series diverges. Full stop.
Think of it as turning a series question into a sequence question. You already know how to take limits of sequences, so {S_n} is the bridge that lets you handle infinite sums with tools you have. Every convergence test in Unit 10 (ratio, comparison, alternating series, all of them) is ultimately a shortcut for answering one question without computing S_n directly. Does the sequence of partial sums settle down to a number?
This is the foundation of Topic 10.1 (Defining Convergent and Divergent Infinite Series) and learning objective 10.1.A, which asks you to determine whether a series converges or diverges. Everything else in Unit 10 is built on top of this definition. Geometric series sums, the nth term test, p-series, even Taylor series in Topics 10.11-10.15 all trace back to "does the sequence of partial sums have a limit?" If you understand this one idea, the rest of the unit stops feeling like a pile of disconnected tests and starts feeling like a toolbox for one job. Unit 10 is BC-only, and it's heavily weighted on the BC exam, so this definition pays rent all the way through the test.
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nth Partial Sum (Unit 10)
The nth partial sum S_n is one entry in the sequence; the sequence of partial sums is the whole list. You also need the reverse direction. Given a formula for S_n, you can recover the general term with a_n = S_n − S_(n−1), which is a classic MCQ move.
Geometric and Telescoping Series (Unit 10)
These are the two famous series types where you can actually write a closed formula for S_n and take its limit directly. Geometric series collapse via the formula a(1−r^n)/(1−r), and telescoping series collapse because middle terms cancel. They're the proof-of-concept that the partial-sum definition really works.
The nth Term Test for Divergence (Unit 10)
If {S_n} converges, then a_n = S_n − S_(n−1) must go to 0 (both partial sums approach the same limit). That's where the nth term test comes from. It's not a separate magic rule; it falls straight out of the partial-sum definition.
Limits of Sequences (Unit 10) and Limits at Infinity (Unit 1)
Deciding convergence of {S_n} is just taking a limit as n→∞, the same skill as horizontal asymptotes from Unit 1. S_n = (2n−1)/n → 2 is the exact same calculation as finding the horizontal asymptote of (2x−1)/x.
This shows up almost entirely as multiple choice, and the questions come in two flavors. Flavor one hands you a formula for S_n and asks what the series does. If S_n = ln(n+1), the partial sums grow without bound, so the series diverges even though its terms shrink. If S_n = (2n−1)/n, the limit is 2, so the series converges to 2. Flavor two runs the other way and asks you to recover the general term using a_n = S_n − S_(n−1). Watch for a sneaky variant where the problem gives a sequence of partial sums that itself looks like series terms, such as S_n = ln(n+1) − ln n; the sum of the series is lim S_n = ln(1) = 0, and if you start telescoping you've confused S_n with a_n. No released FRQ uses the phrase verbatim, but the definition is the justification behind every convergence claim you write in a Unit 10 FRQ.
A series has two sequences attached to it, and mixing them up is the number one Unit 10 error. {a_n} is the list of individual terms; {S_n} is the running total of those terms. Convergence of the series is defined by the limit of {S_n}, not {a_n}. The harmonic series is the cautionary tale. Its terms 1/n go to 0, but its partial sums grow like ln(n) and never settle, so the series diverges. Terms shrinking is necessary for convergence but never sufficient.
The sequence of partial sums {S_n} is the running total of a series, where S_n adds the first n terms (LIM-7.A.1).
A series converges to S if and only if lim(n→∞) S_n = S; if that limit doesn't exist, the series diverges (LIM-7.A.2).
Given a formula for S_n, take its limit to find the sum of the series, and use a_n = S_n − S_(n−1) to recover the general term.
Don't confuse {S_n} with {a_n}; the terms of the harmonic series go to 0, but its partial sums blow up, so it diverges.
S_n = ln(n+1) diverges (grows without bound) while S_n = (2n−1)/n converges to 2, so always actually take the limit instead of eyeballing.
Every convergence test in Unit 10 is a shortcut for the same question: does the sequence of partial sums have a finite limit?
It's the sequence {S_n} where S_n is the sum of the first n terms of a series. The AP definition (LIM-7.A.2) says a series converges to S exactly when lim(n→∞) S_n = S, so this sequence is literally how convergence is defined in Topic 10.1.
No. Terms going to 0 is necessary but not sufficient. The harmonic series Σ1/n has terms that go to 0, but its partial sums grow without bound, so it diverges. Convergence depends on the limit of the partial sums, not the terms.
The sequence of terms {a_n} lists each individual term; the sequence of partial sums {S_n} keeps a running total. For Σ1/2^n, the terms are 1/2, 1/4, 1/8, ... while the partial sums are 1/2, 3/4, 7/8, ... approaching 1. The series sum is the limit of {S_n}, which is 1.
Subtract consecutive partial sums: a_n = S_n − S_(n−1) for n ≥ 2, and a_1 = S_1. So if S_n = (2n−1)/n, compute (2n−1)/n − (2n−3)/(n−1) and simplify. This is a standard multiple-choice setup.
No, it's BC-only. All of Unit 10 (Infinite Sequences and Series) appears only on the BC exam, where it's one of the most heavily tested units, including a routine spot in the free-response section.
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