calc series tests

nth term test / divergence test

If the limit of the sequence as it approaches infinity DNE or does not equal zero, the series diverges

integral test

suppose the associated function f is a continuous, positive, and decreasing function for large values of x

The series will converge iff the improper integral of f converges

p-series test

suppose the series 1/(n^p): the series is convergent for p > 1 and divergent for p <= 1

comparison test

Suppose the series a and b have positive terms

If b is convergent and a <= b for all n, then a is convergent

If b is divergent and a >= b for all n, then a is divergent

limit comparison test

Suppose the series a and b have positive terms

If the limit as b approaches infinity of a/b is a constant, then both series either converge or diverge

(SET B AS LARGEST TERM IN NUMERATOR AND DENOMINATOR)

alternating series test

If an alternating series -1^(n-1) b with b being positive is:

1. decreasing

2. 0 as lim n → infinity

It is convergent

absolute convergence test

suppose a series of absolute values of a is convergent (it is absolutely convergent): the series a is therefore convergent