Calculus 2 Infinite Series & Convergence Tests
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30 Terms
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→∞ (an) is nonzero or doesn’t exist
When is the test for divergence inconclusive?
lim n→∞ (an) = 0
What are the conditions for using the integral test?
after rewriting an as f(x), f must be
continuous
positive
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 Σan converge by the direct comparison test?
Σbn converges and an <= bn
When does a series Σan diverge by the direct comparison test?
Σbn diverges and an >= bn
what can we conclude from the limit comparison test?
If lim n→∞ an / bn = finite number > 0, Σbn and Σan have the same convergence behavior (either both converge or both diverge)
* choose Σbn 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→∞ an / bn = 0 or ∞
When does an alternating series Σ(-1)^n * bn converge by the alternating series test?
bn+1 <= bn ({bn} is decreasing)
lim n→∞ bn = 0
When is the alternating series test inconclusive?
At least one of the following conditions is violated:
bn+1 <= bn ({bn} is decreasing)
lim n→∞ bn = 0
When is a series Σan absolutely convergent?
Σ|an| converges (and therefore Σan converges too)
When is a series Σan conditionally convergent?
Σ|an| diverges and Σan converges
When is a series Σan absolutely convergent (and therefore convergent) by the ratio test?
lim n→∞ | an+1 / an | is a finite number <1
When is a series Σan divergent by the ratio test?
lim n→∞ | an+1 / an | > 1 or is ∞
When is the ratio test inconclusive?
lim n→∞ | an+1 / an | = 1
When is a series Σan absolutely convergent (and therefore convergent) by the root test?
lim n→∞ n√|an| is a finite number <1
When is a series Σan divergent by the root test?
lim n→∞ n√|an| > 1 or is ∞
When is the root test inconclusive?
lim n→∞ n√|an| = 1
When is a sequence {an} convergent?
lim n→∞ an = a finite number
When is a sequence {an} divergent?
lim n→∞ an doesn’t exist (undefined or goes to ∞)
what are the conditions for using the direct comparison test?
all terms of an and bn must be > 0
what are the conditions for using the limit comparison test?
all terms of an and bn 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?
has the form Σ(-1)^n * bn or Σ(-1)^(n+a number) * bn
bn > 0
