The nth partial sum, written S_n = a_1 + a_2 + ... + a_n, is the sum of the first n terms of an infinite series. On AP Calc BC, a series converges to S if and only if the limit of its sequence of partial sums equals S (LIM-7.A.2).
The nth partial sum is exactly what it sounds like. Take an infinite series, stop after n terms, and add up what you have. That running total is S_n = a_1 + a_2 + ... + a_n (essential knowledge LIM-7.A.1). So S_1 is just the first term, S_2 is the first two terms added together, S_3 is the first three, and so on.
Here's why BC cares so much about it. You can't literally add infinitely many numbers, so partial sums are how calculus sneaks up on an infinite sum. As n grows, the partial sums S_1, S_2, S_3, ... form a sequence, and you ask where that sequence is heading. If lim n→∞ S_n exists and equals some real number S, the series converges and its sum is S. If the limit doesn't exist (it blows up or bounces around), the series diverges (LIM-7.A.2). In other words, an infinite sum is really just the limit of finite sums. That one sentence is the foundation of all of Unit 10.
This term lives in Topic 10.1 (Defining Convergent and Divergent Infinite Series) in Unit 10, which is BC-only material. It directly supports learning objective AP Calc 10.1.A, determining whether a series converges or diverges. Every convergence test you learn later in Unit 10 (geometric, p-series, ratio test, alternating series) is really a shortcut for answering one question, which is whether the sequence of partial sums has a limit. If you understand partial sums, the definition of convergence stops feeling like jargon and starts feeling obvious. Series of constants also set up Taylor and power series at the end of Unit 10, where partial sums become the polynomial approximations you actually compute with.
Keep studying AP® Calculus Unit 10-infinite-sequences-and-series-bc-only
Sequence of Partial Sums (Unit 10)
S_1, S_2, S_3, ... is itself a sequence, and the series converges exactly when this sequence does. This is the bridge between sequences and series, so the limit skills from the start of Topic 10.1 carry straight into series questions.
Limits at Infinity (Unit 1)
Evaluating lim n→∞ S_n is the same skill as finding horizontal asymptotes back in Unit 1. If S_n = n²/(n+1), you compare growth rates and see it goes to infinity, so the series diverges. Old limit tricks, new packaging.
Geometric Series (Unit 10)
The geometric series formula S = a/(1-r) comes from taking the limit of the closed-form partial sum. It's the cleanest example of the partial-sum definition actually producing a sum, which is why Topic 10.2 follows immediately after.
Taylor Polynomials (Unit 10)
A Taylor polynomial is literally a partial sum of a Taylor series. When you approximate e^x with a degree-4 polynomial, you're computing S_4 of an infinite series, so this Topic 10.1 idea returns at the end of the unit.
This is BC-only and shows up most often in multiple choice. The classic setups give you a formula for S_n and ask you to do one of three things. First, evaluate it, like finding the sum of the first 3 terms when S_n = n/(n+2) (just plug in n = 3). Second, take lim n→∞ S_n to decide convergence and find the sum, so S_n = ln(n+1) means the series diverges because the limit is infinite. Third, recover the general term using a_n = S_n − S_(n−1), like extracting a_n from S_n = 1 − 1/(n+1). No released FRQ has used the phrase 'nth partial sum' verbatim, but the concept underwrites every series FRQ, and alternating series error-bound questions explicitly compare a partial sum to the true sum.
a_n is a single term of the series; S_n is the running total of the first n terms. Mixing them up is the most common Unit 10 error. If lim a_n = 0, that tells you nothing on its own (the harmonic series has terms going to 0 but diverges). If lim S_n exists, the series converges, full stop. They're connected by a_n = S_n − S_(n−1), so given a formula for one, you can find the other.
The nth partial sum S_n is the sum of the first n terms of a series, so S_3 means add up exactly the first three terms.
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, you can recover individual terms with a_n = S_n − S_(n−1).
Don't confuse lim S_n with lim a_n; the terms going to 0 does not mean the series converges, but the partial sums having a limit always does.
Finding lim n→∞ S_n uses the same techniques as limits at infinity from Unit 1, like comparing degrees of polynomials.
Every convergence test in Unit 10 is a shortcut for answering whether the sequence of partial sums has a limit.
It's S_n = a_1 + a_2 + ... + a_n, the sum of the first n terms of an infinite series. It's defined in Topic 10.1 (LIM-7.A.1) and is how convergence of a series is officially defined on the exam.
No. The harmonic series 1 + 1/2 + 1/3 + ... has terms going to 0 but its partial sums grow without bound, so it diverges. Convergence requires the partial sums S_n to approach a finite limit, not just the terms a_n.
The nth term a_n is one entry in the series; the nth partial sum S_n is the total of the first n entries. They're linked by a_n = S_n − S_(n−1), which exam questions use, like finding a_n when S_n = 1 − 1/(n+1).
Take the limit of S_n as n goes to infinity. For example, if S_n = n²/(n+1), the limit is infinite, so the series diverges; if S_n = 1 − 1/(n+1), the limit is 1, so the series converges to 1.
No, it's BC-only. Partial sums live in Unit 10 (Infinite Sequences and Series), which doesn't appear on the AB exam at all.
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