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Comparison Tests
is a geometric series.
is a p-series.
is easily integrated.
0 2
1
nn
0 2nn
n
02
1
n n
02 1
1
n n
22 3
n
nan
22
2
3
n
nan
To check the convergence of an unknown series, compare it to a series you know.
is not
is not
is not
Direct Comparison Test (DCT)
For series with positive terms, let 0 ≤ an ≤ bn for all n
If converges, then (something smaller) converges
If diverges, then (something larger) diverges
na
na nb
nb
Examples
Does converge or diverge?
1 23
1
nn
Examples
Does converge or diverge?
1 2
ln5
n n
n
Try:
12 23
1
n n
1 54
3
nn
n
1 12
1
nn
Limit Comparison Test (LCT) na nbGiven two series and . Let’s compare the terms
of each using limits.
If , then . 1lim
n
n
n b
ann ba
So if bn converges, an also converges. If bn diverges, an
also diverges.
cb
a
n
n
n
limIf , then .
nn cba So if bn converges, an also converges. If bn diverges, an
also diverges.
The constant doesn’t change the convergence of the series since it can be factored out.
Limit Comparison Test (LCT) na nbGiven two series and . Let’s compare the terms
of each using limits.
If , then . 0lim
n
n
n b
ann ba
So if bn converges, the smaller an also converges. If bn diverges, we don’t know about an .
n
n
n b
alimIf , then .
nn ba If bn diverges, the larger an
also diverges. If bn
converges, we don’t know about an. We are back to the Direct Comparison Test.
Limit Comparison Test (LCT)For series with positive terms,
If a finite, nonzero constant, then the series
behave the same, both diverge or both converge.
n
n
n b
alim
So what about ?
1 12
1
nn
More examples: Converge or Diverge.
12 543
1
n nn
1 23
1
n n
More examples: Converge or Diverge.
135
2
4
10
n nn
n
1 !
1
n n
Practice:p.573 #1-25odd, 27-34