Calculus 2: Series Tests

Test for Divergence

Series Type: ALL

When to try: First

Geometric Series

Series Type: Sum(1..Inf) a*r^(n-1)

Integral

Series type: f(n) is positive, continuous, decreasing

When to try: f(n) must be integrable

P-Series

Series type: Sum(1..Inf) 1/(n^p)

Direct Comparison

Series type: An and Bn are positive

When to try: If it looks close to a p-series or geometric

Limit Comparison

Series type: An and Bn are positive

When to try: if a Direct Comparison Test does not work

Alternating Series

Series type: Sum(1..Inf)(-1^(n-k))*Bn

* Use test for divergence if limit does not equal 0

Absolute convergence

Series type: ALL

When to try: With trig. series or some alternating series

Ratio Test

Series Type: All

When to try: With factorials or exponentials

Series Expansion: 1/(1-x)

Series Expansion: e^x

Series Expansion: Sin(x)

Series Expansion: Cos(x)