Strategies for Convergence/divergence

divergence test

take limit as n approaches infinity of an

if the limit is 0 the test doesn’t work

if it does not equal 0 the series diverges

geometric series

ar^n-1 if the absolute value of r is less than one the series converges to a/1-r

if r is greater than one the series diverges

p-series

1/n^p if p is greater than one it converges if p is less than or equal to one it diverges

comparison test

compare with known series bn

if an< bn and bn converges than an converges

if an>bn and bn diverges than an diverges

Limit comparison test

compare with known series bn by evaluation L the limit as n approaches infinity of an/bn

if L is a real number not equal to zero, then an and bn both converge or both diverge

if L = 0 and bn converges, then an converges

L=infinity and bn diverges then an diverges

Integral Test

if there exists a positive continuous decreasing function f(x) with the same terms as an, take the integral from N to infinity

the integral and the original series both converge or both diverge

Alternating Series

only works for alternating sries

bn+1 has to be less than or equal to bn for all n>1 and bn has to be approaching zero for the series to converge

Ratio Test

evaluate limit as n approaches infinity of the absolute value of an+1/an

if the limit (p) is less than 1 but greater than or equal to zero, the series converges absolutely

if p is greater than one the series diverges

if p=1 the test is inconclusive

Root test

take the limit as n approaches infinity of the nth root of the absolute value of an

if the limit (p) is less than 1 but greater than or equal to zero, the series converges absolutely

if p is greater than one the series diverges

if p=1 the test is inconclusive