Series Tests
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Last updated 9:07 PM on 4/9/23
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11 Terms
1
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Divergence Test
lim an= 0, inconclusive
lim an ≠ 0, diverges
lim an ≠ 0, diverges
2
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Geometric Series
Form: ar^(n-1)
|r| < 1, converges
|r| ≥ 1, diverges
\
ar^(n0)/1-r is where it converges to
|r| < 1, converges
|r| ≥ 1, diverges
\
ar^(n0)/1-r is where it converges to
3
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P - series
Form: 1/n^p
p > 1, converge
p ≤ 1, diverge
p > 1, converge
p ≤ 1, diverge
4
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Telescoping series
Form:
lim Sn n→ infinity = L → converge
lim Sn n → infinity + or - infinity, DNE → diverges
lim Sn n→ infinity = L → converge
lim Sn n → infinity + or - infinity, DNE → diverges
5
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Integral Test
Form: an=f(n)
Requirements:
* positive
* continuous
* decreasing eventually
Integral 1 → infinity f(x)dx = L → converges
Integral 1 → infinity f(x)dx = + or - infinity, DNE → diverges
Requirements:
* positive
* continuous
* decreasing eventually
Integral 1 → infinity f(x)dx = L → converges
Integral 1 → infinity f(x)dx = + or - infinity, DNE → diverges
6
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Ratio Test
Must be b^n or higher
Form: lim n→ infinity |(an+1)/an| (use when n!)
lim an < 1 → converges
lim an > 1/infinity → diverges
lim an = 1, inconclusive
Form: lim n→ infinity |(an+1)/an| (use when n!)
lim an < 1 → converges
lim an > 1/infinity → diverges
lim an = 1, inconclusive
7
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Root test
Must be b^n or higher
Form: nsqrt(|an|) (use when #^n)
lim an < 1 → converges
lim an > 1/infinity → diverges
lim an = 1, inconclusive
Form: nsqrt(|an|) (use when #^n)
lim an < 1 → converges
lim an > 1/infinity → diverges
lim an = 1, inconclusive
8
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Comparison Test
0 ≤ an ≤ bn
If ∑ bn → converges
∑ an → converges
(If the big series converges, the small series converges)
\
If ∑ an → diverges
∑ bn → diverges
(If the small series diverges, the big series diverges)
If ∑ bn → converges
∑ an → converges
(If the big series converges, the small series converges)
\
If ∑ an → diverges
∑ bn → diverges
(If the small series diverges, the big series diverges)
9
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Limit Comparison Test
Very similar to ratio test, only with two terms
Form: lim → an/bn = L → both sides converge/diverge
If ∑ an → converges, ∑ bn converges
If ∑ an → diverges, ∑ bn diverges
Form: lim → an/bn = L → both sides converge/diverge
If ∑ an → converges, ∑ bn converges
If ∑ an → diverges, ∑ bn diverges
10
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Alternating Series Test
Requirements:
* lim n→ infinity an = 0
* a(n+1) (a sub n + 1) ≤ an (decreasing)
If it meets these requirements, it converges
If not, use another test
* lim n→ infinity an = 0
* a(n+1) (a sub n + 1) ≤ an (decreasing)
If it meets these requirements, it converges
If not, use another test
11
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Harmonic series
1/k → diverges