AP Calculus BC - Unit 10 Study Guide
AP Calculus BC • Unit 10

Infinite Sequences & Series

Exam Weight: ~17–18% • Topics 10.1 – 10.11
10.1

Sequence Basics (Converge vs Diverge)

A sequence is a function with input n (a positive integer): an.

Definition

Converges if limn→∞ an = L (a finite number).
Diverges if it doesn't approach a single finite value.

Graph: Convergent vs Divergent sequences
Dots represent terms an at integer n. AP frequently asks you to identify convergence from a graph/table.
High-yield: Use limit laws + dominant term tricks. Example: an = (3n² + 1)/(n² − 5) → 3.
10.2

Series & Sigma Notation

A series adds sequence terms: Σ an.

Partial Sums

Define Sk = a1 + a2 + ... + ak.
The series Σ an converges if limk→∞ Sk exists.

Graph: Partial sums approaching a limit
A series converges when its partial sums settle to a finite value. This is the official definition.
10.3

Geometric Series (The One You MUST Memorize)

Σn=0∞ arn = a/(1−r) (only if |r| < 1)
How to recognize it instantly
  • Each term is multiplied by a constant ratio: r = an+1/an
  • Index shift doesn't matter: start at 1 or 0, still geometric
Score-5 move: Convert repeating decimals to geometric series. Example: 0.777... = 7/10 + 7/100 + 7/1000 + ...
10.4

nth-Term Test for Divergence (Fastest Test)

If limn→∞ an ≠ 0 (or DNE), then Σ an diverges.
But: if the limit equals 0, you still don't know — you need another test.
AP habit: They will try to trick you into thinking "limit 0 ⇒ convergent." That is FALSE. Example: harmonic series Σ 1/n diverges even though 1/n → 0.
10.5

Integral Test (Bridge from Unit 8)

When you can use it

If an = f(n), where f(x) is continuous, positive, and decreasing for x ≥ 1, then:

Σ an and ∫1∞ f(x) dx either both converge or both diverge.
Graph: Integral Test picture (series vs area)
The rectangles represent an and the curve represents f(x). The integral compares "infinite area" to "infinite sum."
10.6

p-Series (Memorize This Rule)

Σ 1/np converges if p > 1, diverges if p ≤ 1
Instant ID: If it looks like 1/np (or can be simplified into it), use p-series.
10.7

Comparison Tests (How AP expects you to argue)

Direct Comparison Test (DCT)

If 0 ≤ an ≤ bn and Σ bn converges → Σ an converges.
If 0 ≤ bn ≤ an and Σ bn diverges → Σ an diverges.

Limit Comparison Test (LCT) (most common on FRQ)

Choose bn (usually p-series) and compute limn→∞ an/bn = c. If 0 < c < ∞, they behave the same (both converge or both diverge).

Score-5 move: Pick bn by keeping only the dominant term(s). Example: (3n² + 1)/(n⁴ + 5) behaves like 3/n².
10.8

Alternating Series (Leibniz Test) + Error

Alternating Series Test (AST)

For Σ (−1)n bn (or (−1)n+1): it converges if (1) bn decreases and (2) lim bn = 0.

Alternating Series Error Bound: If you stop at the Nth term, then the error is at most the next term: |RN| ≤ bN+1.
Graph: Alternating partial sums "bounce" to a limit
Alternating partial sums typically overshoot/undershoot but tighten toward the limit. This is why the next-term error bound works.
10.9

Ratio & Root Tests (The factorial/exponential detectors)

Ratio Test (best when you see factorials)
L = lim |an+1/an|
If L < 1 converge, if L > 1 diverge, if L = 1 inconclusive.
Root Test (best when you see n-th powers)
L = lim n√|an|
If L < 1 converge, if L > 1 diverge, if L = 1 inconclusive.
Quick choice: factorials → Ratio Test. terms like (3n) or (nn) → Root Test.
10.10

Power Series (interval + radius of convergence)

A power series looks like:

Σ cn(x − a)n
What AP wants
  • Find the radius R and interval of convergence.
  • Use Ratio Test on the general term.
  • Always check endpoints separately (they can change!).
Graph: Convergence behavior around the center
A power series converges inside an interval centered at a: (a − R, a + R). Endpoints must be tested.
10.11

Taylor & Maclaurin Series BC

This is the BC "boss level." But the pattern is consistent: memorize the core Maclaurin series and build everything else using algebra + substitution + differentiation/integration.

Core Maclaurin Series (must know)
FunctionSeries
exΣ xn/n!
sin xΣ (−1)n x2n+1/(2n+1)!
cos xΣ (−1)n x2n/(2n)!
1/(1−x)Σ xn (|x| < 1)
ln(1+x)Σ (−1)n+1 xn/n (|x| < 1, x > −1)
arctan xΣ (−1)n x2n+1/(2n+1) (|x| ≤ 1 endpoints special)
Typical FRQ asks: (1) write a series for a new function, (2) radius/interval of convergence, (3) approximate a value, (4) error bound using next term.
Score-5 Pattern: substitution

Want a series for sin(3x)? Replace x with 3x in the sin series.
Want a series for ∫ sin x / x dx? Use series for sin x, divide by x, integrate term-by-term.

Graph: Taylor polynomial approximating sin(x)
Taylor polynomials are accurate near the center (here: 0). AP will ask about "error" and "how many terms."
← Review Unit 9 Unit 10 Quiz →
🎉

Nice work finishing Unit 10!

Ready to test yourself? Create a free account to take Quiz 10!

Sign Up Free & Take Quiz 10 →

Free account = this quiz unlocked. No credit card needed.