Convergence and Divergence Tests

Geometric Series

(a) / (1 - r)

|r| < 1 --> converges

Telescopic Series

Manipulate a fraction into a state where the second fraction is subtracted from the first. Then, write out a series of terms to cancel as many as possible. Then, take the limits of the terms remaining.

Lim of Sn is Finite Number = converges

If Not = diverges

Test for Divergence

If the limit as n goes to infinity is NOT 0, then the series diverges.

If the limit as n goes to infinity IS 0, then we cannot determine divergence or convergence.

Integral Test

If:

(1) f(x) >= 0 for x >= 1

(2) f'(x) is negative (the function is decreasing)

(3) The limit as x approaches infinity is 0

Then, if the integral from 1 to infinity of f(x) converges, then the series converges. If the integral from 1 to infinity of f(x) diverges, then the series diverges.

p-Series Test

1 / n^p converges if p > 1

If p <= 1, then the series diverges

Direct Comparison Test

(1) If 0 <= an <= bn and bn converges, then an also converges.

(2) If 0 <= an <= bn and an diverges, then bn must also diverge.

Limit Comparison Test

L = the limit as n approaches infinity (an / bn):

(1) If 0 < L < infinity, then the two series have the same behavior. They converge or diverge together.

(2) If L = 0 and bn converges, then an also converges.

(3) If L = Infinity and bn diverges, then an diverges.

Alternating Series Test

If:

(1) f'(x) is negative (the function is decreasing)

(2) The limit of bn as n goes to infinity is 0.

Then, the series converges. If not, the series diverges.

Ratio Test

Let L = the limit of |(an+1) / an|

If:

(1) 0 <= L < 1, then an absolutely converges

(2) L > 1, then an diverges

(3) L = 1, then the ratio test is inconclusive

Root Test

Let L = the limit of |an|^(1/n) as n goes to infinity

If:

(1) 0 <= L < 1, then an is absolutely convergent

(2) L > 1, then an diverges

(3) If L = 1, then the test is inconclusive