Series Tests

Divergence Test

lim n → infinity of An =/ 0, the series DIVERGES

Geometric Series

| r | < 1, the series CONVERGES

| r | >= 1, the series DIVERGES

P - Series

p > 1, the series CONVERGES

p <= 1, the series DIVERGES

Integral Test

An = f(n)

f must be +, continuous, and decreasing from [N, infinity)

= finite value L, the series CONVERGS

± infinity or DNE, the series DIVERGES

Ratio Test

lim n → infinity of | An+1/ An |

< 1, the series CONVERGES

>1 or infinity, the series DIVERGES

= 1, inconclusive

Root Test

lim n → infinity of nth root of | An |

< 1, the series CONVERGES

> 1 or infinity, the series DIVERGES

= 1, inconclusive

Direct Comparison Test

0 <= An <= Bn

big series (Bn) converges, the series CONVERGES

Limit Comparison Test

lim n → infinity of An/Bn = L (finite value)

both series will either converge or diverge

Alternating Series Test

(-1)^n An, must pass divergence test first

An+1 must be decreasing (decreasing terms)

Absolute Convergence Test

| An | converges, then An converges and the series CONVERGES ABSOLUTELY

| An | diverges but An converges, the series is CONDITIONALLY CONVERGENT

if both diverge, the series DIVERGES