Series Tests

Nth Term for Divergence

If the limit as n approaches infinity does not equal 0, the series diverges.

Geometric Series Test

If the absolute value of r is less than one, the series converges. If it converges, you can calculate what it converges to by the formula a1/1-r.

If the absolute value of r is greater than one, the series diverges.

P-Series Test

If p is greater than 1, the series converges.

If p is less than or equal to one, the series diverges.

Direct Comparison Test

Compare the series to something that you do know.

If the smaller series diverges, the larger series diverges. If the larger series converges, the smaller series converges.

Limit Comparison Test

Compare the series to one of the same degree ratio. Then, divide it by the original function. If this equals a positive number, they either both converge or both diverge.

Integral Test

Test that the function is positive, continuous, and decreasing.

If so, if the integral of the series converges, the series will converge. If the integral of the series diverges, the series will diverge. (we love improper integrals)

Alternating Series

If the limit as n approaches infinity equals 0, and the function decreasing, the series converges.

Ratio Test

Best for exponentials and factorials!

If the absolute value of the limit of avn+1/avn is less than one, the series converges. If the absolute value of the limit of avn+1/avn is greater than one, the series diverges.

Root

If the nth root of the series is greater than one, the series converges. If the nth root of the series less than one, the series diverges.