Convergence Tests - MATH 1400

Nth term test

if the limit = 0 the test is inconclusive, if the limit ≠ 0 then the test diverges.

Direct comparison test

Both series must be greater than 0, if b_n is greater than a_n then they both converge. If a_n is greater than b_n then they both diverge.

Limit comparison test

do a_n/b_n and find out what about the a_n transfers. if one (con/di)verges the other (con/di)verges

Ratio test

a_(n+1)/a(n). 1< L diverges, 1>L converges, L = 1 inconclusive.

Integral test

if int from 1 to infinity converges then a(n) converges. f(n) must be positive, continuous, and decreasing.

Alternating Series

(-1)^n(b_n). b_n must be decreasing and lim as n —> infinity 0 to converge.

Absolute and Conditional Convergences

if |a_n| converges then a_n converges absolutely. If a_n converges but |a_n| diverges, it converges conditionally