Calculus 2 Infinite Series & Convergence Tests

When is a geometric series convergent?

|r|<1

When is a geometric series divergent?

|r|>=1

What is the sum of a geometric series?

a/(1-r) , where a is the first term

* can only find the sum if the geometric series is convergent

When is a p series convergent?

p > 1

When is a p series divergent?

p <= 1

When is a series divergent by the test for divergence?

lim n→∞ (a_n) is nonzero or doesn’t exist

When is the test for divergence inconclusive?

lim n→∞ (a_n) = 0

What are the conditions for using the integral test?

after rewriting a_n as f(x), f must be

1. continuous

2. positive

3. decreasing

on [n, ∞)

When does a series converge by the integral test?

∫_n∞ f(x) dx converges

(lim t→∞ ∫_{n ^t} f(x) dx is a constant)

When does a series diverge by the integral test?

∫_n∞ f(x) dx diverges

(lim t→∞ ∫_{n ^t} f(x) dx is undefined or goes to infinity)

When does a series Σa_n converge by the direct comparison test?

Σb_n converges and a_n <= b_n

When does a series Σa_n diverge by the direct comparison test?

Σb_n diverges and a_n >= b_n

what can we conclude from the limit comparison test?

If lim n→∞ a_n / b_n = finite number > 0, Σb_n and Σa_n have the same convergence behavior (either both converge or both diverge)

* choose Σb_n to be a simpler series whose convergence behavior we can figure out, usually a p series or geometric series

when is the limit comparison test inconclusive?

lim n→∞ a_n / b_n = 0 or ∞

When does an alternating series Σ(-1)^n * b_n converge by the alternating series test?

1. b_n+1 <= b_n ({b_n} is decreasing)

2. lim n→∞ b_n = 0

When is the alternating series test inconclusive?

At least one of the following conditions is violated:

1. b_n+1 <= b_n ({b_n} is decreasing)

2. lim n→∞ b_n = 0

When is a series Σa_n absolutely convergent?

Σ|a_n| converges (and therefore Σa_n converges too)

When is a series Σa_n conditionally convergent?

Σ|a_n| diverges and Σa_n converges

When is a series Σa_n absolutely convergent (and therefore convergent) by the ratio test?

lim n→∞ | a_n+1 / a_n | is a finite number <1

When is a series Σa_n divergent by the ratio test?

lim n→∞ | a_n+1 / a_n | > 1 or is ∞

When is the ratio test inconclusive?

lim n→∞ | a_n+1 / a_n | = 1

When is a series Σa_n absolutely convergent (and therefore convergent) by the root test?

lim n→∞ ⁿ√|a_n| is a finite number <1

When is a series Σa_n divergent by the root test?

lim n→∞ ⁿ√|a_n| > 1 or is ∞

When is the root test inconclusive?

lim n→∞ ⁿ√|a_n| = 1

When is a sequence {a_n} convergent?

lim n→∞ a_n = a finite number

When is a sequence {a_n} divergent?

lim n→∞ a_n doesn’t exist (undefined or goes to ∞)

what are the conditions for using the direct comparison test?

all terms of a_n and b_n must be > 0

what are the conditions for using the limit comparison test?

all terms of a_n and b_n must be > 0

what condition must be met to use the ratio test?

all terms of the series are nonzero

what conditions must be met to use the alternating series test?

1. has the form Σ(-1)^n * b_n or Σ(-1)^(n+a number) * b_n

2. b_n > 0