Convergence/Divergence Tests

Divergence Test

If n does not = 0 as it approaches infinity, it DIVERGES

Geometric Series Test: How to Identify

= ar^n (Anytime there is an exponent look to pull a common ratio)

Using Geometric Series Test

If 0< |r| < 1 = CONVERGES

|r| ≥ 1 = DIVERGES

P-series Test: How to Identify

1/n^p

Using P-series Test

If p > 1 = CONVERGES

If p ≤ 1 = DIVERGES

Comparison Test: How to Identify

If it is complicated but similarly structured to a simpler series

Examples of series to use for comparison

1/n = diverges, 1/n! = absolutely converges, 1/n(n+1) = Converges

Using the Comparison Test

MUST BE 0 ≤ aₙ ≤ bₙ for all n

If Σbₙ converges, then Σaₙ also converges. If Σbₙ diverges, then Σaₙ also diverges.

Limit Comparison Test: How to Identify

If it is too complicated to compare using comparison test, use a series that you know diverges/converges

Using the Limit Comparison Test

1. Take the limit as n approaches infinity of the ratio of the two series.

2. If the limit is a positive finite number, both series converge/diverge together.

Alternating Series Test: How to Identify

If the series is alternating in sign for each n

Using Alternating Series Test

If an is positive, decreasing, and approaches 0 as n → infinity and a n+1 ≤an​ for all n, then the series converges.

Ratio Test: How to identify

Good for getting rid of factorials (!)

Using Ratio test

1. Take the limit of the absolute value of a(n+1)/a(n).

2. If the limit is less than 1, the series converges.

3. If the limit is greater than 1 or infinite, the series diverges.

4. If the limit is 1, the test is inconclusive.

Root Test: How to identify

Terms to the nth power with no factorials

Using the Root Test

limⁿ→∞ |an|^(1/n). If the limit < 1, the series converges absolutely; if > 1, it diverges; if = 1, the test is inconclusive.