convergence test

test for divergence (tfd)

L doesn’t equal 0

tfd diverges

geometric series test

geometric series converges

geometric series diverges

p-series test

p-series converges

p-series diverges

integral test

positive, continuous, decreasing

integral test conditions

integral test converges

finite integral

integral test diverges

-infinity and +infinity

direct comparison test (dct)

dct converges

dct diverges

limit comparison test (lct)

lct converges

p > 1

lct diverges

p < 1

ratio test

ratio test converges

L > 1

ratio test diverges

L < 1 or

L = infinity

ratio test inconclusive

L = 1

root test

root test converges

L < 1

root test diverges

L > 1 or

L = infinity

root test inconclusive 

L = 1

alternating series test (ast)

ast converges

terms are decreasing, limit = 0