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18 Cards in this Set
- Front
- Back
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The divergence test |
This test can give us two results.. if lim an != 0, then diverges if lim an = 0, inconclusive |
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The integral test |
The integral test will usually follow divergence test if the divergence test comes up inconclusive.
if an goes to f(n), and f(n) is positive decreasing and continuous, then the summation a(n) is equal to definite integral for f(n), and either both converge or diverge |
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the ratio test |
If a(n) is a series with positive terms, then
r= lim (a(n+1))/a(n)... If r < 1, series converges If r> 1 or lim a(n+1)/a(n) >inf, then diverges If r = 1, then test is inconclusive |
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the p-series test |
The p-series test states that for 1/n^p...
if p > 1 then series converges else if p <= 1 , series diverges |
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geometric series test |
In a geometric series, for a^n...
if abs(a) < 1, then the series converges to a/1-r if abs(r) >= 1, then series diverges |
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telescoping |
You're gonna do this!
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Key Limits: lim (1+k/n)^n |
e^k |
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key Limit: lim n^(1/n) = lim n√n |
1
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key Limit: lim n!/n^n AND n^n/n! |
0 AND diverges infinity
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key Limit: r^n |
if abs(r) < 1= 0; if r= 1 =1; otherwise DNE/diverges; |
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For geometric series, if a and r are hard to find then we can do... |
write out first few terms, where first one is a, and r is what we need to multiply a(n) by to get a(n+1)
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L'hopitals: 0/0 ; 0 * Inf |
For 0/0, we just take derivative of top and bottom SEPERATELY...
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L'HOPITALS: 1^inf ; inf^0 ; 0^0 |
Here, we can bring exponent down, making it a 0*inf, then carry out that process... a^b = b ln(a) = ln(a)/(1/b) = ln Y, so we must remember answer will really by e^y (solving for y) |
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L'HOPITALS: inf - inf |
turn into one term = lim a/b, then react from there. |
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two problems of improper integrals ∫ |
Two kinds of problems, one where one of the end points goes to inf, and one where point between the two goes to inf. |
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two fixes for improper integrals |
When integral goes from a to inf, switch inf to b and then do lim(b--> inf). When some point between two goes to inf, SEE NOTES |
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10!/8! = (4! *2!)/(3!) = |
= 9*10 =4 * 2! = 8 |
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increasing sequence/ non-decreasing: decreasing/ non increasing: bounded : monotone: |
increasing or decreasing is when a(n+1) is >/< a(n), while non-increasing/non-decreasing is <=/>= a(n)...
bounded has to do with the what the max-y/min-y of any function is... |
