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Limit Comparison k>0
Ratio Test k=1
Alternating Series requirements
positive decreasing and lim=0
How do you find what a sequence converges to?
Take the limit of the sequence as n approaches infinity
What is a geometric sequence?
Each term is multiplied by a common ratio
What are the two sequences within a series?
sequence of terms and sequence of partial sums
When does a sequence converge?
If and only if its terms approach some real number as n approaches infinity.
When does a series converge?
If and only if its sequence of partial sums converges.
How do you calculate the sum of the first n terms of a geometric series?
How do you calculate the sum of an infinite geometric series?
When can you calculate the sum of an infinite geometric series?
when the absolute value of the common ratio is less than one
Name all the tests we know for series convergence
geometric series where common ratio<1; integral test; p-series; ratio test; order comparison test (OCT); alternating series test (AST)
cannot be determined by the ratio test
In the ratio test, if the limit as n approaches infinity is=1, the series...
State the Order comparison test in words.
Convergence of the dominant forces convergence of the subdominant. Divergence of the subdominant forces divergence of the dominant.
State the nth term test in words
if a series converges, then the limit of the series as n approaches infinity is 0.
If the limit of a series as n approaches infinity is 0, does that mean the series converges?
Name all the tests we know for divergence.
nth term test; integral test; harmonic series relative; p-series; order comparison test
continuous, positive-valued, decreasing
What condition(s) are necessary for the integral test?
If the integral test shows convergence, do the series and the integral converge to the same number?
What is required of a function to be considered a harmonic series relative?
must take the form 1/n where n is linear and has a positive slope
Can an alternating series be considered a harmonic series relative?
do truncation and rearrangement affect what a series converges to?
infinite series (a) + infinite series (b)
a represents a function. b represents a second function. What is the infinite series of (a+b) equal to?
k * infinite series of (a)
a represents a function. What is the infinite series of k*a equal to?
Which is larger: n! or 2^(n-1)?
Which is larger (except one case where they are equal): ln(n) or n?
true or false: tan^-1(n) is always ≤ π/2?
What condition(s) are necessary for the order comparison test?
exponentials or factorials
When is the ratio test particularly helpful?
When will the ratio test always be inconclusive?
What is an alternating series?
terms must follow the pattern +/-/+/-/+/- or -/+/-/+/-/+
What condition(s) are necessary for the Alternating Series Test (AST)?
alternating, limit as n approaches infinity is 0, strictly decreasing
Does the Alternating Series Test prove absolute convergence?
the n+1st term
If you use the nth partial sum to approximate S (the sum of the entire series), your error will always be strictly less than...
lim an ≠ 0
Diverges by nth term test when
|r| < 1
Converges by Geometric Series Test when
|r| > 1
Diverges by Geometric Series Test when
p > 1
Converges by p-series test when
p ≤ 1
Diverges by p-series test when
Converges by Ratio test when
lim ( |an+1| / |an| ) < 1
Diverges by Ratio test when
lim ( |an+1| / |an| ) > 1
less than conv.
Converges by Direct Comparison test when
greater than diver.
Diverges by direct comparison test when
Converges by limit comparison when
lim ( |bn| / |an|) = +# & bn converges
Diverges by limit comparison when
lim ( |bn| / |an| ) = +# & bn diverges
Converges by Alternating series test when
lim an = 0 & an+1 < an
Converges by nth term test when
∫conv. => ∑conv.
Converges by integral test when
∫div. => ∑div.
Diverges by integral test when
Sum of a geometric series
S = (a₁ / 1−r)