AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/10/10-1)
Unit 10 · Topic 10.1
BC only10.1 Defining Convergent and Divergent Infinite Series
BC only. An infinite series adds up infinitely many terms. You decide what that means by looking at the partial sums S₁, S₂, S₃, …: if they approach a finite number S, the series converges to S, and if they don't, the series diverges.
Key terms
- sequence
- series
- partial sum
- converge
- diverge
Sequences and series
This whole unit is BC only. A sequence is an ordered list of numbers: a₁, a₂, a₃, … . A series is what you get when you add the terms of a sequence: a₁ + a₂ + a₃ + …, written Σ (n = 1 to ∞) aₙ. The letter n is the index; it may start at 0, 1 or any other whole number.
Keep the two ideas separate. The sequence 1/2, 1/4, 1/8, … approaches 0. The series 1/2 + 1/4 + 1/8 + … adds up to 1.
Partial sums
You can't literally add infinitely many numbers, so you add the first n terms and see what happens as n grows. The nth partial sum is Sₙ = a₁ + a₂ + … + aₙ. The partial sums form a new sequence: S₁, S₂, S₃, … .
If lim (n → ∞) Sₙ = S for a finite number S, the series converges, and its sum is S. If the limit is infinite or doesn't exist, the series diverges. A divergent series has no sum.
- Converges: 1/2 + 1/4 + 1/8 + … has Sₙ = 1 − 1/2ⁿ, which approaches 1.
- Diverges to infinity: 1 + 2 + 3 + … has partial sums that grow without bound.
- Diverges by bouncing: 1 − 1 + 1 − 1 + … has partial sums 1, 0, 1, 0, …, which never settle.
Going back and forth between terms and partial sums
If you know the partial sums, you can recover each term: aₙ = Sₙ − Sₙ₋₁ for n ≥ 2, and a₁ = S₁. If you have a formula for Sₙ, its limit tells you whether the series converges.
Changing or removing finitely many terms never changes whether a series converges. It can change the sum, though. So the starting index matters for the value but not for convergence.
What the rest of the unit does
For most series you can't find a formula for Sₙ. The rest of this unit gives tests (geometric, nth term, integral, p-series, comparison, alternating series, ratio) that decide convergence without computing partial sums. Then it uses series to build polynomials and power series that represent functions.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
From partial sums to terms and sum
The nth partial sum of a series Σ (n = 1 to ∞) aₙ is Sₙ = n/(n + 1). (a) Find a₅. (b) Does the series converge? If so, to what?
Show the solutionHide the solution
- Step 1: (a) a₅ = S₅ − S₄ = 5/6 − 4/5 = 25/30 − 24/30 = 1/30.
- Step 2: (b) lim (n → ∞) n/(n + 1) = 1. The partial sums approach 1, so the series converges to 1.
Answer: (a) a₅ = 1/30 (b) It converges to 1.
- Example 2
A sum that collapses
Find the sum of Σ (n = 1 to ∞) 1/(n(n + 1)).
Show the solutionHide the solution
- Step 1: Split each term: 1/(n(n + 1)) = 1/n − 1/(n + 1). (Check: 1/n − 1/(n + 1) = (n + 1 − n)/(n(n + 1)).)
- Step 2: Write out a partial sum: Sₙ = (1 − 1/2) + (1/2 − 1/3) + … + (1/n − 1/(n + 1)).
- Step 3: Everything in the middle cancels, leaving Sₙ = 1 − 1/(n + 1).
- Step 4: lim (n → ∞) Sₙ = 1. This is called a telescoping series.
Answer: The series converges to 1.
- Example 3
Trap: terms approach 0 doesn't mean the series converges
The sequence aₙ = 1/n approaches 0. Does Σ (n = 1 to ∞) 1/n converge?
Show the solutionHide the solution
- Step 1: Terms going to 0 is about the sequence, not the series.
- Step 2: Group the terms: 1 + 1/2 + (1/3 + 1/4) + (1/5 + … + 1/8) + … . Each group in parentheses is at least 1/2 (for example, 1/3 + 1/4 > 1/4 + 1/4).
- Step 3: There are infinitely many groups, each adding at least 1/2, so the partial sums grow without bound.
Answer: No. The harmonic series diverges, even though its terms go to 0.
Common mistakes
- Mixing up the limit of the terms aₙ with the limit of the partial sums Sₙ.
- Saying a series converges because its terms get small.
- Using aₙ = Sₙ − Sₙ₋₁ and then subtracting in the wrong order.
- Assuming the starting index doesn't affect the sum. It doesn't affect convergence, but it does affect the value.
On the exam
- Multiple-choice questions may give a formula for Sₙ and ask for the sum or for a particular term.
- In free response, when you conclude a series converges or diverges, name the test or the reason. “Converges” alone earns little.
Connected topics
Videos
Check yourself
4 questions on 10.1 Defining Convergent and Divergent Infinite Series. Pick an answer to see if you got it, and why.
The nth partial sum of the series Σ (n = 1 to ∞) aₙ is Sₙ = 4n/(2n + 1). Which of the following is true?
What is the sum of the series Σ (n = 1 to ∞) (1/(n + 1) − 1/(n + 2))?
The nth partial sum of the series Σ (n = 1 to ∞) aₙ is Sₙ = 3 − 2/n. What is the value of a₅?
Let aₙ = 2n/(n + 5) for n ≥ 1, and let Sₙ = a₁ + a₂ + ⋯ + aₙ. Which of the following statements are true? I. The sequence {aₙ} converges. II. The series Σ (n = 1 to ∞) aₙ converges. III. The sequence of partial sums {Sₙ} is unbounded.
0 of 4 answered