Calculus 2: Series Tests for Convergence and Divergence

nth Term Test for Divergence

If lim(n→∞) aₙ ≠ 0, then ∑aₙ diverges.

Geometric Series

What are the conditions for convergence of a geometric series?

|r| < 1 --> series converges

What are the conditions for divergence of a geometric series?

|r| ≥ 1 --> series diverges

P-Series Test

Conditions of Convergence: p > 1

Conditions of Divergence: p ≤ 1

Telescoping Series Test

Conditions of Convergence: lim (n-->∞) Sn = L

Conditions of Divergence: lim (n--> ∞)Sn = +/-∞/DNE

Alternating Series Test

Not used for divergence

Conditions for Convergence: 0 < an+1 ≤ an and lim (n-->∞) an = 0

Integral Test ( f must be continuous, positive, and decreasing)

Condition for convergence: ∫f(x)dx = L

Condition for divergence: ∫f(x)dx = +/-∞/DNE

Ratio Test

Conditions for convergence: lim | an + 1/ an | < 1

Conditions for divergence: lim |an + 1/an| > 1/∞

=1 --> inconclusive

Root Test

Direct Comparison Test

If An < Bn, then An converges if Bn converges; If An < Bn, then Bn diverges if An diverges

Limit Comparison Test

if the limit as n approaches infinity of (given series/chosen series) is >0, the two series will converge/diverge together.