---
title: "Sequence of Partial Sums — AP Calc BC Definition & Guide"
description: "The sequence of partial sums {Sn} adds the first n terms of a series; a series converges exactly when this sequence has a limit. Core to AP Calc BC Unit 10."
canonical: "https://fiveable.me/ap-calc/key-terms/sequence-of-partial-sums"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 10-infinite-sequences-and-series-bc-only"
---

# Sequence of Partial Sums — AP Calc BC Definition & Guide

## Definition

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.

## What It Is

Take any [infinite series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB "fv-autolink") 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](/ap-calc/key-terms/nth-partial-sum "fv-autolink")** (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](/ap-calc/unit-1/determining-limits-using-algebraic-properties-limits/study-guide/HjStgVKViPGZj1CxYwEB "fv-autolink") 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?

## Why It Matters

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](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/working-with-geometric-series/study-guide/YvDdN4qbMyMaLgnFApU9 "fv-autolink") or [diverges](/ap-calc/key-terms/diverges "fv-autolink"). 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.

## Connections

### [nth Partial Sum (Unit 10)](/ap-calc/key-terms/nth-partial-sum)

The nth partial sum S_n is one entry in the [sequence](/ap-calc/key-terms/sequence "fv-autolink"); 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](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/nth-term-test-for-divergence/study-guide/oEMEbEp7gWXCgxDFyRlx "fv-autolink") 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](/ap-calc/unit-1 "fv-autolink"). S_n = (2n−1)/n → 2 is the exact same calculation as finding the horizontal asymptote of (2x−1)/x.

## On the AP Exam

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.

## sequence of partial sums vs sequence of terms {a_n}

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.

## Key Takeaways

- 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?

## FAQs

### What is the sequence of partial sums in AP Calc BC?

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.

### If the terms of a series go to 0, does the series converge?

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.

### How is the sequence of partial sums different from the sequence of 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.

### How do I find a_n if I'm given a formula for S_n?

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.

### Is the sequence of partial sums on the AP Calculus AB exam?

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.

## Related Study Guides

- [10.1 Defining Convergent and Divergent Infinite Series](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/defining-convergent-divergent-infinite-series/study-guide/CIVFHStGQM90EJ4GtIDB)

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