Sect. 9-D Comparison Tests
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Transcript of Sect. 9-D Comparison Tests
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SECT. 9-D COMPARISON TESTS
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The Direct comparison Test If 0 < an < bn for all n and positive terms, then:
If the larger series converges, then the smaller series must also converge.
If the smaller series diverges then the larger series must also diverge
converges a then converges, if1 1n
n
nnb
diverges b then diverges, if1 1n
n
nna
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The Direct comparison Test Given a seriesCompare with a similar Geometric or p- Series
When choosing a series for comparison you can disregard all but the highest power in the numerator and denominator
When choosing an appropriate p-series, you must choose one with the same nth term
1 1n2
n
3 2 with 14
2 comparen
n
nnn
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1. Does converge or diverge?
12 342
5n nn
12
12 2
5 ? 342
5nn nnn
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2. Does converge or diverge?
1 231
nn
11 21 ?
231
nn
nn
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3. Does converge or diverge?
122
1n n
12
12
1 ? 44
1nn nnn
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4. Does converge or diverge?
1 11
n n
11
1 ? 1
1nn nn
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The Limit comparison Test If an > 0, and bn > 0 for all n , then:
Where L is finite and positive, then both converge or they both diverge
Works well when comparing a messy algebraic series to a p-series or geometric series
Lba
n
n
n
lim
11
and n
nn
n ba
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5. Does converge or diverge?
15
2
5
32n n
nn
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6. Does converge or diverge?
123 54n nn
n
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7. Does converge or diverge?
12
)ln(n n
n
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