Math 152 – Spring 2016
Section 10.3
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Section 10.3 – The Integral and Comparison Tests,
Estimating Sums
Example 1. Use the integral of the corresponding function to determine if the following
series are convergent or divergent.
(a)
∞
X
1
2
n
n=1
(b)
∞
X
1
√
n
n=1
The Integral Test. Suppose that f is a function on [1, ∞) and let an = f (n). If f is
• continuous,
• positive, and
• decreasing,
then
Z
∞
1. If
f (x) dx is convergent, then
1
Z
2. If
an is convergent.
n=1
∞
f (x) dx is divergent, then
1
∞
X
∞
X
an is divergent.
n=1
Note.
• The Integral Test still works if the series starts at something other than
n = 1. If the series starts at n = 2, then start the integral at 2 as well.
• We must make sure the function f satisfies all three requirements or the integral
test does not apply. However, it is not necessary that f is always decreasing, just
that f is only decreasing after some point (i.e., there is some N such that f is
decreasing for all x > N ).
Math 152 – Spring 2016
Section 10.3
Example 2. Determine whether the series
∞
X
ln n
is convergent or divergent.
n2
n=1
Example 3. For what values of p is the series
P-Series: The p-series
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∞
X
1
convergent?
np
n=1
∞
X
1
is convergent if p > 1 and divergent if p ≤ 1.
p
n
n=1
Note: It is easy to confuse p-series and geometric series:
∞
X
1
: the index n is part of the base n1 and p is the power (or exponent).
p
n
n=1
The series converges based on how big the exponent p is: converges if p > 1,
diverges if p ≤ 1.
• p-series,
• Geometric series,
∞
X
arn−1 . The index n is part of the exponent, and the con-
n=1
vergence of the series depends on the base r. Converges if |r| < 1, and diverges if
|r| ≥ 1.
Math 152 – Spring 2016
Example 4. Determine if the series
Section 10.3
∞
X
3 of 6
3n−3 is convergent or divergent.
n=1
P
P
The Comparison Test. Suppose that
an and
bn are series with positive terms
(i.e., an > 0 and bn > 0 for all n).
X
P
bn is convergent and an ≤ bn for all n, then
an is convergent.
• If
• If
X
bn is divergent and an ≥ bn for all n, then
P
an is divergent.
Example 5. Determine if the following series are convergent or divergent.
(a)
∞
X
8n + 3
4n2 − 3n
n=1
(b)
∞
X
5n − 4
32n + 2n
n=1
Math 152 – Spring 2016
Section 10.3
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P
P
The Limit Comparison Test: Suppose that
an and
bn are series with positve
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 6. Determine if the following series are convergent or divergent.
(a)
∞
X
8n − 3
4n2 + 3n
n=1
∞
X
1
(b)
sin
n
n=1
Hint: lim
x→0
sin x
= 1 (use L’Hospital or see p.181)
x
Math 152 – Spring 2016
Section 10.3
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Question. Suppose we know that a series converges by the integral test, but we can
not calculate its sum. How do we approximate the sum and give a margin of error for
our approximation?
Remainer Estimate for the Integral Test: If
and Rn = s − sn , then
Z ∞
Z
f (x) dx ≤ Rn ≤
n+1
P
an converges by the Integral Test
∞
f (x) dx
n
Example 7. (a) Approximate the sum of the series
∞
X
1
by using the sum of the
n3
n=1
first 10 terms.
(b) How many terms for the previous series are required to ensure that the sum is
accurate to within 0.0005?
Math 152 – Spring 2016
Section 10.3
Example 8. How many terms of the series
6 of 6
∞
X
1
n=1 n(1 + ln n)
5
would you need to find
the sum is accurate to within 0.001?
Question. Does the previous theorem on remainders give a better estimate of the sum?
What if we add sn to the above inequality?
Theorem. If
P
an converges by the Integral Test, Rn = s − sn , and
Z ∞
Z ∞
sn +
f (x) dx ≤ s ≤ sn +
f (x) dx
n+1
P
an = s, then
n
Example 9. Use this theorem with n = 10 to estimate the sum of the series
∞
X
1
.
3
n
n=1