c Dr Oksana Shatalov, Fall 2012
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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?
np
n=1
∞
X
1
is called harmonic series.
If p = 1 then
n
n=1
y
x
0
THE INTEGRAL TEST Suppose f is a continuous, positive, decreasing function on [1, ∞) and
Z ∞
∞
X
an is convergent if and only if the improper integral
f (x) dx
let an = f (n). Then the series
n=1
is convergent. In other words:
FACT: The p-series,
∞
X
1
, 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
1
c Dr Oksana Shatalov, Fall 2012
(b)
∞
X
n=2
2
1
n ln n
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
4n
1
+ n4
c Dr Oksana Shatalov, Fall 2012
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THE COMPARISON TEST Suppose that
an ≤ bn for all n.
• If
P
bn is convergent then
• If
P
an is divergent then
P
P
P
an and
P
bn are series with positive terms and
an is also convergent.
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
n2001 − sin2000 n
∞
X
cos4 n
√
(c)
n2 n
n=1
c Dr Oksana Shatalov, Fall 2012
(d)
∞
X
5n + 1
4n
n=1
(e)
4
∞
X
n=0
4n
1
+ n4
Warning: Distinguish between
∞
X
n
p
and
∞
X
pn .
n=1
n=1
In some cases inequalities are useless. For example, for the series
∞
X
n=0
we have
4n
1
−n
c Dr Oksana Shatalov, Fall 2012
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THE LIMIT COMPARISON TEST Suppose that
P
an and
P
bn are series with positive terms .
If
an
=c
bn
where c is a finite number and c > 0, then either both series converge or both diverge.
lim
n→∞
EXAMPLE 3. Determine if the following series is convergent or divergent:
(a)
∞
X
n=1
4n
1
−n
∞
X
n2 + n
√
(b)
n5 + n3
n=2
c Dr Oksana Shatalov, Fall 2012
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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, Fall 2012
∞
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?
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c Dr Oksana Shatalov, Fall 2012
EXAMPLE 5. Given
8
∞
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,