AP Calculus BC Unit 10 Convergence & Divergence Tests

nth-Term Test

Diverges if the limit does not equal zero

Geometric Series

Does an = ar^n-1, n >/= 1

If not, use a different test

If |r| < 1 then convergent

If |r| > 1 then Divergent

Example : Σ5(1/2)^n

r = 1/2 which is less than 1, so it is convergent

P-Series

p>1 converges

p<1 diverges

Example : an = 1/n^P

Alternating Series

Does an = (-1)^n bn, or

an = (-1)^n+1 bn, b > 0

If yes, then is limbn = 0 and bn+1/bn < 1

Or is the nth term approaching 0 and is the term decreasing. If so, it is convergent

Example : Σ(-1)(5x)^2/(n^3)+8

Approaches 0 and is decreasing so it is convergent

Comparison Test

Let 0 ≤ an ≤ bn, for all n.

1) If ∑bn converges, then ∑an converges.

2) If ∑an diverges, then ∑bn diverges.

Limit Comparison Test

if the limit as n approaches infinity of (given series/chosen series) is >0, the two series will converge/diverge together.

Integral Test

Where f(x) continuous, positive, decreasing over [1, ∞) and an=f(n), if 1∫∞f(x) convergent, Σan convergent. If 1∫∞f(x) divergent, Σan divergent.

Ratio Test

lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges