Tests for Convergence and Divergence of Series

Test for Divergence (limit not 0 or does not exist)

Diverge

Test for Divergence (limit is 0)

Inconclusive

Ratio Test (L < 1)

Converge

Ratio Test (L > 1)

Diverge

Ratio Test (L = 1)

Inconclusive

Root Test (L < 1)

Converge

Root Test (L > 1)

Diverge

Root Test (L = 1)

Inconclusive

p-Series Test (p > 1)

Converge

p-Series Test (p <= 1)

Diverge

Comparison Test (b_n converges and a_n <= b_n)

Converge

Comparison Test (a_n diverges and a_n >= b_n)

Diverge

Limit Comparison Test (limit of (a_n / b_n) > 0)

Both converge or both diverge

Alternating Series Test (b_n decreasing and limit of b_n = 0)

Converge

Integral Test (integral from 1 to infinity converges)

Converge

Integral Test (integral from 1 to infinity diverges)

Diverge

Geometric Series (|r| < 1)

Converge

Geometric Series (|r| >= 1)

Diverge

nth-Term Test (limit of a_n not 0)

Diverge

nth-Term Test (limit of a_n is 0)

Inconclusive