Infinite Series’ Tests

nth Term Test for Divergence

Diverges: if limit is NOT equal to zero

Converges: Can’t be proven with this test

Geometric Series Test

Diverges: if |r| >/= 1

Converges: if |r| < 1

P-Series Test

Diverges: if p > 1

Converges: if 0 < p </= 1

Integral Test

Diverges: if integral from 1-infinity = infinity

Converges: if integral from 1-infinity = finite number

Direct Comparison Test

Compare to a simple series, then use nth term, geometric, p-series, or integral tests to prove.

Limit Comparison Test

Set up a limit ratio of compared function to original function. Evaluate the limit, if it is between 0 and infinity, use nth term, geometric series, p-series, or integral test on the compared function to determine convergence or divergence.

Ratio Test

Diverge: if the ration of the limits is less than 1

Converge: if the ration of of the limits is greater than 1

Root Test

Diverge: if the limit is less than 1

Converge: if the limit is greater than 1