AP Calculus BC Unit 10 Cheat Sheet: Infinite Sequences & Series

The basics

What it covers: Convergence and divergence, the full set of series tests, absolute vs. conditional convergence, Taylor and Maclaurin series, power series, and error bounds. This is a BC-only unit.

Exam weight: About 17–18% of the BC exam — the largest single unit.

The big question: When does an infinite sum have a value, and how do series represent functions?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

Convergence Basics

A series converges when its partial sums approach a finite limit. nth term test: if aₙ ⋮→ 0, it diverges.

Geometric & p-Series

Geometric Σarⁿ converges iff |r| < 1, sum a/(1−r). p-series Σ1/nᵖ converges iff p > 1 (harmonic p = 1 diverges).

Integral & Comparison

Integral test: series & ∫₁∞ f agree. Comparison / limit comparison: bound or compare to a known series.

Alternating & Ratio

Alternating test: decreasing terms → 0. Ratio test: L = lim|aₙ₊₁/aₙ|; L < 1 converges, L > 1 diverges, L = 1 inconclusive.

Absolute vs. Conditional

Absolute: Σ|aₙ| converges (⇒ converges). Conditional: Σaₙ converges but Σ|aₙ| diverges.

Error Bounds

Alternating: |error| ≤ |first omitted term|. Lagrange: |Rₙ| ≤ (M/(n+1)!)|x−a|ⁿ⁺¹.

Taylor / Maclaurin

Pₙ(x) = Σ f⁽ᵏ⁾(a)/k!·(x−a)ᵏ. Maclaurin: a = 0. Know eˣ, sin x, cos x, 1/(1−x).

Power Series

Radius R from the ratio test; converges for |x−a| < R. Test endpoints for the interval of convergence.

The key tests and formulas you must know

Convergence — partial sums approach a finite limit.
Geometric series — Σarⁿ = a/(1−r) for |r| < 1.
p-series — Σ1/nᵖ converges iff p > 1.
nth term test — aₙ ⋮→ 0 ⇒ diverges.
Integral / comparison / limit comparison tests.
Alternating series test — decreasing terms → 0.
Ratio test — L = lim|aₙ₊₁/aₙ|.
Absolute vs. conditional convergence.
Alternating error bound — ≤ first omitted term.
Lagrange error bound — (M/(n+1)!)|x−a|ⁿ⁺¹.
Taylor / Maclaurin series — Σ f⁽ᵏ⁾(a)/k!·(x−a)ᵏ.
Radius / interval of convergence — from the ratio test + endpoints.

Key themes to remember

Pick the right test. Recognize the form: geometric, p-series, alternating, ratio-friendly.
aₙ → 0 is necessary, not sufficient. The harmonic series proves it.
Absolute convergence is stronger. It guarantees plain convergence.
Series represent functions. Taylor/Maclaurin build them; error bounds control accuracy.
Reuse known series. Differentiate, integrate, or substitute into eˣ, sin x, 1/(1−x).

Common exam traps

nth term test only shows divergence. aₙ → 0 proves nothing by itself.
Ratio test L = 1 is inconclusive — try another test.
Always test the endpoints of a power series’ interval separately.
p > 1 converges, p ≤ 1 diverges — the boundary p = 1 (harmonic) diverges.
Alternating error ≤ first omitted term, only for a valid alternating series.
Lagrange needs a bound M on the (n+1)th derivative over the interval.
Don’t forget the factorial in Taylor coefficients: f⁽ᵏ⁾(a)/k!.