In AP Calculus BC, the nth partial sum Sₙ is the sum of the first n terms of a series (EK LIM-7.A.1). A series converges to a sum S if and only if the limit of its sequence of partial sums exists and equals S (EK LIM-7.A.2).
A partial sum is exactly what it sounds like. Take an infinite series a₁ + a₂ + a₃ + ⋯ and stop after n terms. That running total, Sₙ = a₁ + a₂ + ⋯ + aₙ, is the nth partial sum. So S₁ = a₁, S₂ = a₁ + a₂, S₃ = a₁ + a₂ + a₃, and so on. Each partial sum is just a regular finite number you could compute by hand.
Here's why this idea carries Unit 10 on its back. You can't literally add infinitely many numbers, so the CED defines an infinite series' sum through partial sums instead. The partial sums S₁, S₂, S₃, … form a sequence, and the series converges to S if and only if that sequence of partial sums approaches S as n → ∞. In other words, an infinite sum is really a limit in disguise. Every convergence test you learn later in Unit 10 is ultimately answering one question. Does the sequence of partial sums settle down to a finite number, or not?
Partial sums live in Topic 10.1 (Defining Convergent and Divergent Infinite Series) in Unit 10, which is BC-only. They directly support learning objective 10.1.A, determining whether a series converges or diverges. The two essential knowledge statements (LIM-7.A.1 and LIM-7.A.2) are basically the definition of partial sums plus the definition of convergence built on top of them. That makes this term the foundation the entire unit stands on. Geometric series, the nth term test, telescoping series, even the error bounds for alternating series all trace back to the behavior of Sₙ as n grows. If you understand that 'the series converges' means 'lim Sₙ exists,' the rest of Unit 10 makes a lot more sense.
Keep studying AP Calculus Unit 10
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view galleryInfinite Series (Unit 10)
An infinite series is defined through its partial sums. The 'sum' of a series isn't an actual infinite addition; it's the limit of the sequence S₁, S₂, S₃, …. Partial sums are the bridge between something you can compute (a finite sum) and something you can't (an infinite one).
Convergence (Unit 10)
A series converges to S exactly when lim n→∞ Sₙ = S. That's the official CED definition (LIM-7.A.2), and it's why questions sometimes hand you a formula for Sₙ and ask for the sum of the series. You just take the limit.
Divergence (Unit 10)
If the sequence of partial sums has no finite limit (it blows up, like Sₙ = ln(n+1), or it bounces around), the series diverges. Divergence isn't a separate idea; it's just what happens when the partial sums fail to settle.
Limits of Sequences (Unit 10 / limit ideas from Unit 1)
Evaluating lim Sₙ uses the same limit skills you built back in Unit 1, just applied to a sequence instead of a function. For example, lim n→∞ n/(n+2) = 1 by comparing leading terms, so a series with that partial sum formula converges to 1.
Partial sum questions show up as multiple choice in a few predictable flavors. One type gives you a sequence like aₙ = 1/n³ and asks for S₃, which is pure arithmetic (1 + 1/8 + 1/27). A sneakier type hands you the formula for Sₙ itself, like Sₙ = n/(n+2) or Sₙ = ln(n+1), and asks for the sum of the first few terms, whether the series converges, or the value of lim n→∞ Sₙ. The trap in that second type is treating Sₙ as if it were aₙ. If Sₙ = n/(n+2), the sum of the first 3 terms is just S₃ = 3/5, no adding required. You should also be able to recover individual terms using aₙ = Sₙ − Sₙ₋₁. On free response, partial sums power telescoping series arguments and any justification that a series converges because its partial sums approach a limit.
aₙ is one term of the series; Sₙ is the running total of the first n terms. They're connected by Sₙ = a₁ + ⋯ + aₙ and aₙ = Sₙ − Sₙ₋₁, but they behave very differently. For a convergent series, aₙ must go to 0 while Sₙ goes to the actual sum S. Mixing these up is the single most common partial-sum mistake, especially when a problem gives you a formula for Sₙ and you start summing it like it's aₙ.
The nth partial sum Sₙ is the sum of the first n terms of a series, so S₃ = a₁ + a₂ + a₃.
A series converges to S if and only if the limit of its sequence of partial sums exists and equals S; that is the CED's official definition of convergence (LIM-7.A.2).
If a problem gives you a formula for Sₙ, the sum of the first n terms is just Sₙ plugged in, not a new sum you compute term by term.
You can recover individual terms from partial sums using aₙ = Sₙ − Sₙ₋₁.
If lim n→∞ Sₙ doesn't exist or is infinite (like Sₙ = ln(n+1), which grows without bound), the series diverges.
Partial sums are BC-only content, living in Unit 10 under Topic 10.1.
The nth partial sum Sₙ is the sum of the first n terms of a series. For aₙ = 1/n³, the partial sum S₃ = 1 + 1/8 + 1/27. It's the key tool for defining whether an infinite series converges (Topic 10.1, EK LIM-7.A.1).
No. The nth term aₙ is a single term, while the partial sum Sₙ is the total of the first n terms. They're related by aₙ = Sₙ − Sₙ₋₁, and confusing them is the most common error on these questions.
Take the limit of Sₙ as n → ∞. For example, if Sₙ = n/(n+2), the series converges to 1 because lim n→∞ n/(n+2) = 1. If the limit doesn't exist or is infinite, the series diverges.
Not necessarily. Convergence is about the partial sums having a finite limit, not just the terms shrinking. The harmonic series has terms going to 0, but its partial sums grow without bound, so it diverges.
No. Partial sums are part of Unit 10 (Infinite Sequences and Series), which is BC-only content. On the BC exam they appear mostly in multiple choice, either computing Sₙ directly or analyzing a given Sₙ formula.