Convergence and Divergence Tests for Series

Flashcards summarizing key concepts related to convergence and divergence tests for series.

Geometric Series

Converges if |r| < 1 and diverges if |r| ≥ 1; sum is s = a / (1 - r).

Test for Divergence

A series diverges if lim(n→∞) an ≠ 0 or does not exist; inconclusive if lim(n→∞) an = 0.

p-Series

Converges if p > 1 and diverges if p ≤ 1.

Integral Test

Converges if the integral ∫ from 1 to ∞ of f(x)dx converges; f must be continuous, positive, and decreasing.

Comparison Test

If Σbn converges and an ≤ bn for all n, then Σan converges; if Σbn diverges and an ≥ bn, then Σan diverges.

Limit Comparison Test

If lim(n→∞) an/bn = c (where c > 0), then both series converge or both diverge.

Alternating Series

Converges if bn+1 ≤ bn for all n and lim(n→∞) bn = 0.

Absolute Convergence

A series converges absolutely if Σ|an| converges, implying Σan converges.

Conditional Convergence

A series converges conditionally if Σan converges but Σ|an| diverges.

Ratio Test

Converges absolutely if L < 1, diverges if L > 1 or ∞, inconclusive if L = 1.

Root Test

Converges absolutely if L < 1, diverges if L > 1 or ∞, inconclusive if L = 1.