Convergence and Divergence Tests for Series

Flashcards summarizing key convergence and divergence tests for series.

Test for Divergence

Diverges if limn→∞ an ≠ 0 or limn→∞ an does not exist; inconclusive if limn→∞ an = 0.

Geometric Series

Converges if |r| < 1; diverges if |r| ≥ 1; sum is Σ∞ n=1 arn−1 = a / (1 - r).

p-series

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

Integral Test

Converges if ∫∞1 f(x)dx converges; diverges if ∫∞1 f(x)dx diverges, where f derived from an is continuous, positive, and decreasing.

Comparison Test

If Σbn converges and an ≤ bn for all n ≥ 1, then Σan converges; if Σbn diverges and an ≥ bn for all n ≥ 1, then Σan diverges.

Limit Comparison Test

If limn→∞ an/bn = c > 0, then either both series converge or both diverge.

Alternating Series

Converges if bn+1 ≤ bn for all n ≥ 1 and limn→∞ bn = 0.

Absolutely Convergence

If Σ|an| converges, then Σan converges.

Conditionally Convergence

If Σan converges but Σ|an| diverges.

Ratio Test

If limn→∞ |an+1/an| = L, then the series is absolutely convergent if L < 1, divergent if L > 1 or ∞, inconclusive if L = 1.

Root Test

If limn→∞ n√|an| = L, then the series is absolutely convergent if L < 1, divergent if L > 1 or ∞, inconclusive if L = 1.