Calc 2 Convergence Tests

Integral Test

let the series be a function that is positive, decreasing, and continuous:

• if the limit of the sequence as x approaches infinity converges then the series converges

• if the limit of the sequence as x approaches infinity diverges then the series diverges

p-series

• if the p > 1 the series converges

• diverges otherwise

direct comparison test

• to show that a series converges we must compare it to a larger convergent series

• to show that a series diverges we must compare it to a smaller divergent series

limit comparison test

let a and b be 2 series with positive terms and limit of a/b = L:

• if the limit > 0 and finite, then either both a and b converge or both diverge

• if limit = inf and a converges then b converges

• lf the limit = 0 and b convertes then a converges

geometric series

• if the ratio < 1 the series converges to cr^k/1-r

• otherwise the series diverges

Telescoping Series

find a formula for the Nth partial sum and the limit to infinity

Alternating series test

if:

• b > 0 for all n >= 1

• bn+1 < bn

• lim bn = 0

  the series converges

Ratio test