Every Test for Infinite Series Convergence/Divergence

Chapter 9: Infinite Series

nth term test

condition(s) of convergence: N/A

condition(s) of divergence: the limit as n approaches infinity of the sequence (a_n) is not equal to 0

this test cannot be used to show convergence

geometric series test

condition(s) of convergence: 0 < |r| < 1

condition(s) of divergence: |r| >= 1

sum of the series: S = a/(1 - r)

telescoping series test

condition(s) of convergence: the limit as n approaches infinity of the sequence (a_n) is equal to L

condition(s) of divergence: N/A

sum of the series: S = a₁ - L

p-series test

condition(s) of convergence: p > 1

condition(s) of divergence: 0 < p <= 1

alternating series test

condition(s) of convergence: the series must be alternating AND 0 < a_{n + 1} <= a_n AND the limit as n approaches infinity of the sequence (a_n) is equal to 0

condition(s) of divergence: N/A

remainder: |R_n| <= a_n+1

integral test

condition(s) of convergence: the integral from 1 to infinity of f(x)dx converges

condition(s) of divergence: the integral from 1 to infinity of f(x)dx diverges

note: must also meet the standards of a_n = f(n) = f(x) being positive AND continuous

remainder: 0 < R_n < the integral from n to infinity of f(x)dx

carreon note: his least favorite because it’s really complicated

root test

condition(s) of convergence: the limit as n approaches infinity of the nth-root of the absolute value of the sequence (|a_n|) is less than 1

condition(s) of divergence: the limit as n approaches infinity of the nth-root of the absolute value of the sequence (|a_n|) is greater than 1 (or = infinity)

inconclusive when the limit as n approaches infinity of the nth-root of the absolute value of the sequence (|a_n|) is equal to 1

carreon note: only use this when the entire series it raised to the nth power

ratio test

condition(s) of absolute convergence: the limit as n approaches infinity of the absolute value of the following term over the previous term (|a_{n + 1}/a_n|) is less than 1

condition(s) of divergence: the limit as n approaches infinity of the absolute value of the following term over the previous term (|a_{n + 1}/a_n|) is greater than 1

inconclusive when the limit as n approaches infinity of the absolute value of the following term over the previous term (|a_{n + 1}/a_n|) is equal to 1

carreon note: if the series contains a factorial, use this test

direct comparison test

condition(s) of convergence: 0 < a_n <= b_n AND the infinite series of b_n converges

condition(s) of divergence: 0 < b_n <= a_n AND the infinite series of b_n diverges

limit comparison test

condition(s) of convergence: the limit as n approaches infinity of a_n/b_n is equal to L AND the infinite series of b_n converges

condition(s) of divergence: the limit as n approaches infinity of a_n/b_n is equal to L AND the infinite series of b_n diverges

note: L must be positive and finite

carreon note: his favorite (as opposed to its counterpart) because it has fewer steps