AP Calc BC Unit 10

Partial Sum

Sn = Sn-1 + an

Geometric Sequence

an = a0r^n

Geometric Infinite Series Convergence

When |r| >= 1 a geometric series diverges.

When |r| < 1 the series converges to ar^k / 1-r

nth Term Test for Divergence

If limit as n approaches infinity of an is NOT equal to 0, then an diverges.

Integral Test for Convergence

If f is a positive, continuous and decreasing function for x >=k, and an = f(x), then an and intergral from k to infinity f(x) both converge OR diverge.

P-series

1/n^p

Series converges if p > 1

Series diverges if 0 < p =< 1

Harmonic Series

1/n

Diverges

Comparison Test for Convergence

0 < an =< bn for all n

If bn converges, then an converges.

If bn diverges, then an diverges.

Limit Comparison Test

If an > 0 and bn > 0, and limit as n approaches infinity an/bn = L (L is finite and positive) then an and bn either BOTH converge or diverge.

Alternating Series Test

If an > 0, then the alternating series (-1^nan and -1^n+1an) converge IF:

1. limit as n approaches infinity of an = 0

2. an+1 <= an for all n