Infinite Series: every key term you need

Sequences

Ordered list of numbers a₁, a₂, a₃, ... Converges if lim(n→∞) aₙ = L. Diverges otherwise.

Series Definition

Sum of sequence terms: S = Σ aₙ. Partial sum: S_N = a₁ + a₂ + ... + aₙ.

Geometric Series

Σ ar^(n-1) = a/(1-r) if |r|<1. Convergent if |r|<1, divergent otherwise.

Arithmetic Series

Σ (first term + common diff·n). Diverges unless common diff = 0.

Divergence Test

If lim aₙ ≠ 0, series diverges. If lim aₙ = 0, test inconclusive (series may converge/diverge).

Integral Test

If ∫f(x)dx converges, Σf(n) converges. If integral diverges, series diverges. f must be positive, decreasing.

p-Series

Σ 1/n^p converges if p>1, diverges if p≤1. Example: Σ 1/n² converges, Σ 1/n diverges.

Comparison Test

If 0 < aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges. If 0 < cₙ ≤ aₙ and Σcₙ diverges, then Σaₙ diverges.

Alternating Series

Σ (-1)^n aₙ. Converges if aₙ decreasing and lim aₙ = 0 (alternating series test).

Absolute vs Conditional Convergence

Absolute: Σ|aₙ| converges. Conditional: Σaₙ converges but Σ|aₙ| doesn't.

Power Series

Σ aₙ(x-c)^n. Converges for |x-c| < R (radius of convergence). Found using ratio test.

Unit 7 Summary

Series: sum of sequence. Tests for convergence: divergence, integral, p-series, comparison, alternating. Power series in terms of x.

More AP Calculus BC guides

• Limits & Continuity
• Applications of Derivatives
• Parametric & Polar
• Differential Equations