Calculus BC => Tests for Convergence and Divergence

Geometric Series Test→ what type of series is it for?

Geometric Series Test→ divergent conditions

|r| ≥ 1

Geometric Series Test→ convergent conditions

|r| < 1 and will converge to S= a (first term) / 1-r

nth term test for divergence

image → If =0, it doesn’t tell us anything; need to use another test. can only be used to prove divergence, not convergence

Telescoping Series Test→ conditions for convergence

If in expanded form, the terms begin to subtract out'

When is telescoping test used?

if a series collapses to a finite sum then it converges by this test

What must a function be in order to use integral test?

continuous, positive, decreasing

What kind of limit do you need to set up for integral test

How to find lower bound on limit of series with integral test?

use the answer you found when doing the integral test

How to find upper bound?

integral+lower bound

P-series Test→ form?

P-series test→ what happens for p=1

harmonic series

P-series test→ p>1 means?

converges

P-series test→ 0<p<1 means?

diverges

P-series test→ p=1 means?

harmonic series

Direct Comparison test→ when can you use it?

Direct Comparison Test→ conditions for convergence?

Direct Comparison Test→ conditions for divergence?

Limit Comparison Test→ what kind of function would we see when doing this test?

Limit Comparison Test→ what should be true to conduct this test

Limit Comparison Test→ The two series for a.sub.n and b.sub.n. should?

either both converge or both diverge

Alternating series test→ what should be true about a.sub.n to use test

should be greater than 0 and decreasing

Alternating series test→ what kind of series?

(-1)^n+1 × 1/n & (-1)^n * 1/n! => alternatively positive and negative

Alternating series test→ converges if?

• alternate in sign

• decrease in magnitude

• limit of 0

what is alternate series test used for

ONLY to prove CONVERGENCE

Alternate series test→ if that first limit with a.sub.n doesn’t equal zero?

series diverges by nth term test for divergence NOT by this testA

Alternate series test→ a series is absolutely convergent if?

Alternate series test→ a series is conditionally convergent if?

ratio test→