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When you see natural logs, which test should you try?
the integral test
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1/e (1/e)^n-1
known convergent geometric series
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1/n
known divergent P-series
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1/n^5/3
known convergent P-series
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1/√n
known divergent P-series
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When does a geometric series converge?
when r is between -1<r<1
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When does a P-series converge?
when P>1
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What is the formula for determining what a geometric series converges to?
a/1-r
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What tests should you try when it says not to use ratio or convergence test?
use limit comparison test
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What does the divergence theorem say?
If the limit of the series does not equal 0, the series diverges
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What does the direct comparison test say?
If xn ≤yn then if yn converges, so does xn
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What does the limit comparison test say?
If xn and yn are both infinite series whose terms are all positive, if yn converges and the limit of xn/yn = K where K is a non-negative number, then xn also converges
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range for arctan
(-pi/2, pi/2)
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range for arcsin
[-pi/2, pi/2]
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When using the limit comparison test, how do you know if the series converges?
the limit goes to 0
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Ratio Test Definition
|a (n+1)/a(n)|
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Taylor Series Formula
Σ[f^(n)a]/n! (x-a)^n
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Maclaurin Series
Σf^(n)(0)/n! (x^n)
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What does it mean if you get 0 when doing the ratio test?
the series converges absolutely
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What does it mean when n=0?
that it is most likely a power series (use ratio test)
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Rules for the alternating series test
- 1. Un>0 for all n≥N
- 2. lim Un = 0
- 3. U (n+1)≤ Un for all n≥N
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What test do you use if rule 2 of the alternating series test fails?
the divergence test
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What does it mean when the ratio test = 1?
it doesn't tell us anything
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What does the ratio test tell us if L <1
the series is absolutely convergent
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What does the ratio test tell us when L >1?
the series diverges
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What test should you do if the nth term is algebraic?
the limit comparison test
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