Start with the first 2 sections of the Unit 10 one-page cheatsheet below. The full sheet is a PDF you can print.
Infinite series ∑aₙ converges to S if lim(n→∞) Sₙ = S, where Sₙ = a₁ + a₂ + ... + aₙ (nth partial sum).
Geometric series: ∑(n=0 to ∞) arⁿ = a/(1−r) when |r| < 1; diverges if |r| ≥ 1.
nth Term Test (Divergence Only): If lim(n→∞) aₙ ≠ 0, series diverges. If lim(n→∞) aₙ = 0, test is inconclusive.
Key trap: aₙ → 0 does NOT guarantee convergence.
Integral Test: If aₙ = f(n) where f is positive, continuous, and decreasing on [1, ∞), then ∑_{n=1}^∞ aₙ and the improper integral ∫₁^∞ f(x) dx both converge or both diverge (the integral's value is not the sum).
p-Series: ∑(1/nᵖ) converges if p > 1; diverges if p ≤ 1.
Harmonic series: ∑(1/n) diverges (p = 1).
Alternating harmonic: ∑(−1)ⁿ⁺¹/n converges (by alternating series test).
7 more sections are on the one-page PDF:
Comparison Tests
Alternating Series & Ratio Test
Power Series & Radius of Convergence
Constructing Power Series
Absolute vs. Conditional Convergence
Error Bounds for Approximations
Taylor & Maclaurin Series
Made by Fiveable
Unit 10: Infinite Sequences and Series (BC only) on one page, aligned to the 2026-2027 course. Made and checked by the Fiveable Content Team.
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