c Dr Oksana Shatalov, Spring 2011
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10.3: The Integral and Comparison Tests; Estimating Sums
∞
X
1
QUESTION: For what values of p the p-series
is convergent?
p
n
n=1
∞
X
1
If p = 1 then
is called harmonic series.
n
n=1
y
x
0
y
x
0
c Dr Oksana Shatalov, Spring 2011
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THE INTEGRAL TEST Suppose f is a continuous, positive, decreasing function
∞
X
on [1, ∞) and let an = f (n). Then the series
an is convergent if and only if
n=1
Z ∞
the improper integral
f (x) dx is convergent. In other words:
1
∞
X
1
FACT: The p-series,
, converges if p > 1 and diverges if p ≤ 1.
p
n
n=1
EXAMPLE 1. Determine if the following series is convergent or divergent:
∞
X
1000
√
(a)
n
n
n=1
(b)
∞
X
n=2
1
n ln n
c Dr Oksana Shatalov, Spring 2011
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A disadvantage of the Integral Test: it does force us to do improper integrals
which are in some cases may be impossible to determine the convergence of. For
example, consider
∞
X
n=0
1
4n + n4
c Dr Oksana Shatalov, Spring 2011
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P
P
THE COMPARISON TEST Suppose that
an and
bn are series with positive
terms and an ≤ bn for all n.
P
P
• If
bn is convergent then
an is also convergent.
P
P
• If
an is divergent then
bn is also divergent.
EXAMPLE 2. Determine if the following series is convergent or divergent:
(a)
∞
X
n4 + 4
n=1
(b)
∞
X
n=2
n6 + 6
n2000
n2011 − sin2000 n
∞
X
cos4 n
√
(c)
2 n
n
n=1
c Dr Oksana Shatalov, Spring 2011
(d)
5
∞
X
5n + 1
n=1
4n
Warning: Distinguish between
∞
X
p
n
and
n=1
∞
X
pn .
n=1
In some cases inequalities are useless. For example, for the series
∞
X
n=0
we have
1
4n − n
c Dr Oksana Shatalov, Spring 2011
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P
P
THE LIMIT COMPARISON TEST Suppose that
an and
bn are series with
positive terms . If
an
lim
=c
n→∞ bn
where c is a finite number and c > 0, then either both series converge or both
diverge.
EXAMPLE 3. Determine if the following series is convergent or divergent:
(a)
∞
X
n=1
1
4n − n
∞
X
n2 + n
√
(b)
n5 + n3
n=2
c Dr Oksana Shatalov, Spring 2011
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Illustration: Why c in the Limit Comparison Test must be positive and finite:
REMAINDER ESTIMATE FOR THE INTEGRAL TEST
P
If
an converges by the Integral Test and Rn = s − sn , then
Z ∞
Z ∞
f (x) dx ≤ Rn ≤
f (x) dx
n+1
which implies
Z
n
∞
Z
f (x) dx ≤ s ≤ sn +
sn +
n+1
∞
f (x) dx
n
c Dr Oksana Shatalov, Spring 2011
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∞
X
1
EXAMPLE 4. Given
n3
n=1
(a) Approximate the sum of the series by using the sum of the first 10 terms.
(b) Estimate the error.
(c) How many terms are required to ensure that the sum is accurate to within
0.0005?
c Dr Oksana Shatalov, Spring 2011
EXAMPLE 5. Given
9
∞
X
1 + sin n
n=1
2n3
.
(a) Prove the convergence.
(b) By comparison the series to a p-series, estimate the error in using s100 to
approximate the sum of the series,