Math 1B
§ 11.4 The Comparison Tests
∞
Consider:
1
∑n
converges (p-series, p = 2 > 1)
2
n=1
∞
1
Now consider: ∑ 2
n +1
n=1
€
€
1
1
< 2
n +1 n
2
€
⇒
€
€
∞
1
The partial sums of ∑ 2
are increasing (positive terms) and bounded from above (by
n=1 n + 1
∞
∑n
n=1
∞
1
∑n
2
), so
n=1
1
converges.
+1
€
2
∞
On the other hand,
€
1
∑n
∞
diverges (harmonic series) and
n=1
n=1
∞
its partial sums are larger than those of
€
1
1
>
1
n− 2 n
1
2
1
∑ n , which blow up. So
n=1
€
€
∞
1
∑n−
is term by term greater than
1
∑n
so
n=1
∞
1
∑n−
n=1
1
2
diverges.
€
€
⇒
€
€
Stewart – 7e
1
The Comparison Test: Suppose that
(eventually). Then
i) If
∑b
ii) If
∑a
n
and
∑b
n
are series with positive terms and an ≤ bn
is convergent then ∑ an is convergent.
€
€
n
n
is divergent then
€
€
∑a
∑b
n
€
is divergent.
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Note: To use this test we must have a known series for comparison. We often use one of these:
€
1
• p-series:
∑ n p converges if p > 1 and diverges if p ≤ 1
∞
• Geometric series:
∑ ar
n=1
€
Example: Is the series
€
€
∞
converges if r < 1 and diverges if r ≥ 1
€
n −1
convergent
2
€ or divergent?
n
∑n
n=1
n−1
€
€
Stewart – 7e
2
∞
Example: Is the series
1+ sin n
convergent or divergent?
10 n
n= 0
∑
€
∞
Example: Is the series
ln n
convergent or divergent?
n=1 n
∑
€
Stewart – 7e
3
The Comparison Test can be difficult because you need to “know” the answer (converge or diverge)
and then find a series to compare with in order to prove it.
∞
It fails to be useful for series like
∑n
n= 2
wrong way:
1
where the answer seems obvious but the inequalities go the
−1
2
∞
∞
1
1
>
∑ n 2 −1 ∑ n 2
n= 2
€ n= 2
The Limit Comparison Test: Suppose
€
an
=c
n →∞ b
n
lim
∑a
n
and
∑b
n
are series with positive terms. If
(c is finite, c > 0 )
€
€
then the series either both converge or both diverge.
€
€
∞
Example: Is the series
∑n
n= 2
1
convergent or divergent?
−1
2
€
Stewart – 7e
4
∞
Example: Is the series
3n 3 − 2n 2 + 4
convergent or divergent?
7
3
n=1 n − n + 2
∑
€
∞
Example: Is the series
∑
n=1
4n 2 + n
n5 + n3
convergent or divergent?
€
Stewart – 7e
5
∞
Example: Is the series
4n
convergent or divergent?
n
+1
∑3
n= 0
€
Stewart – 7e
6