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22 Cards in this Set
- Front
- Back
- 3rd side (hint)
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Give an example of a sequence
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An=1/n
1, 1/2, 1/3, 1/4, 1/5... |
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Give an example of a series
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Summation E1/n
1+1/2+1/3+1/4+1/5... |
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What is the harmonic series, and does it converge or diverge?
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S=Ʃ1+1/2+1/3+1/4+1/5...
It diverges |
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If Ʃan and Ʃbn converge then:
Ʃ(an+bn)=? ƩCan=? |
Ʃan + Ʃbn
CƩan |
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An arithmetic series always _____ unless zero
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Will always diverge unless 0+0+0+0...
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What is a geometric sequence?
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a, ar, ar^2, ar^3...
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what is a geometric series?
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a+ar+ar^2+ar^3...
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What is an arithmetic series?
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a, a1 + d, a1 + 2d, a1 + 3d...
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What is a partial sum?
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Sn=a+ar+ar^2+ar^3+...ar^n-1
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"Telescoping Factor"
Sn=(a+ar+ar^2+ar^3+...ar^n-1) |
(1-r)Sn=(a+ar+ar^2+ar^3+...ar^n-1)(1-r)
(1-r)Sn= a+(-ar)+ar+(-ar^2)+ar^2+...ar^n-1+(-ar^n) |
(1-r)Sn=a-ar^n=a(1-r^n)
Sn=(a(1-r^n))/(1-r) |
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When does a geometric series converge?
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Ʃar^n converges if and only if IrI<1 and will converge to ?
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Converges to ar^k/(1-r)
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Ʃan
an----> C C does not = 0 |
Series diverges via DT
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Divergence Test:
Given a series Ʃan if an does not go to zero then the series what? |
The series diverges
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What does the Integral test tell you have a series Ʃan that is continuous non-negative decreasing function f(x) where f(n)=an?
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Ʃan and ʃf(x)dx
either both converge or both diverge |
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How do you know if a series diverges using P-integral?
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If P>1 then the series converges.
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Given: Ʃan and Ʃbn
0<an<bn |
1) if Ʃbn converges then Ʃan converges
2) if Ʃan diverges then Ʃbn diverges |
:)
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Limit Comparison Test
Given 2 series Ʃan and Ʃbn lim(an/bn) |
if 0<lim(an/bn)<∞ then they either both converge or both diverge
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Given an alternating series:
Ʃ((-1)^n)an , an is >/= 0 |
1) lim an=0
2) an is decreasing (derivative < 0) |
Then the series converges
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Facts
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If ƩIanI converges then Ʃan converges absolutely
If ƩIanI diverges but Ʃan converges then Ʃan converges conditionally |
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Ratio Test: Given series Ʃan , an any real number
consider limIan+1/anI=L |
1) if L<1 series converges absolutely
2) if L>1 series diverges 3) if L=1 test fails |
Example: Ʃ1/n diverges
lim(1/n+1)*n/1=n/n+1=1 test fails |
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Definition:
Limit of a Sequence |
Let L be a real #, the limit of a sequence {an} is L, written as Liman=L
if for each €>0, there exists M>0 such that Ian-LI<€ whenever n>M |
If the limit L of a sequence exists, then the sequence converges to L. If the limit of a sequence does not exist, then the sequence diverges.
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Theorem 9.1 Limit of a Sequence
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Let L be a real #. Let f be a function of a real variable such that limf(x)=L
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If {an} is a sequence such that f(n)=an for every positive integer n, then liman=L
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