Section 10.3: The Integral and Comparison Tests
The Integral Test:
If f (x) is a positive, continuous, decreasing function on [1, ∞], and ak = f (k).
Z
Then
∞
P
∞
ak and
k=1
f (x) dx either both converge or both diverge. Why is this so?
1
EXAMPLE 1: Determine whether the following series converge or diverge.
a.)
b.)
n
n=1 n + 1
∞
P
∞
P
n=1
c.)
∞
P
n2
n
+1
ne−n
2
n=0
d.)
1
n=2 n ln n
∞
P
EXAMPLE 2: For what value(s) of p does the series
1
converge?
p
n=1 n
∞
P
1
1
1
= 1 + p + p + ..., where p > 0. The
p
2
3
n=1 n
special case where p = 1 is called the harmonic series:
P-series Test: A p-series is a series of the form
The p series test says
∞
P
1
is convergent if p > 1 and divergent if p ≤ 1.
n=1 np
∞
P
EXAMPLE 3: Determine whether the series
1000
√ converges or diverges.
n=1 n n
EXAMPLE 4: Determine whether the series
3
√ converges or diverges.
n
n=1
Note that if
P
∞
P
an is similar to a p series, then we may want to compare the series to a p series.
The Comparison Test: Suppose
∞
P
n=1
• If
∞
P
an and
∞
P
bn are series of positive terms.
i=1
∞
P
bn is convergent and an ≤ bn for all n, then
∞
P
bn is divergent and an ≥ bn for all n, then
n=1
the larger series converges, so does the smaller series”.
• If
n=1
smaller series diverges, so does the larger series”.
∞
P
n=1
∞
P
an is also convergent. In other words, ”if
an is also divergent. In other words, ”if the
n=1
Note: if the larger series diverges, no conclusion can be made about the smaller series. Likewise, if
the smaller series converges no conclusion can be made about the larger series.
EXAMPLE 5: Determine whether the following series converge or diverge.
∞
n4
P
a.)
8
2
n=1 n + n + 1
b.)
∞
P
n=17
√
n2
n5 − 4n
c.)
sin2 n
√
n=1 n n
d.)
cos2 n + 5
√
3
n
n=1 n +
e.)
5 + sin n
√
n
n=1
∞
P
∞
P
∞
P
The Limit Comparison Test: If the Comparison Test is inconclusive, then we may apply the
Limit Comparison Test:
Suppose
∞
P
an and
∞
P
bn are series of positive terms.
n=1
n=1
an
= c > 0, then either both series converge or both diverge.
bn
an
an
= 0 or lim
= ∞, then the test fails and therefore we need to apply another
Note: If lim
n→∞ b
n→∞ b
n
n
test.
If n→∞
lim
EXAMPLE 6: Determine whether the following series converge or diverge.
∞ n2 − 5n
P
a.)
n=1 n3 + 2n
1
sin 2
b.)
n
n=1
∞
P
c.)
sin
∞
P
cos
n=1
d.)
n=1
1
n
∞
P
1
n
Remainder Estimate: Suppose sn =
n
P
ai = a1 + a2 + ... + an is the nth partial sum of the
i=1
convergent series
∞
P
n=1
an . Then the remainder in using sn to approximate the sum S is defined to
n=1
∞
P
be Rn = S − sn =
Moreover, if
∞
P
ai = an+1 + an+2 + ....
i=n+1
an was shown to be convergent by the integral test where an = f (n), then
Rn =
∞
X
i=n+1
ai <
Z
∞
n
f (x) dx.
1
2
n=1 n + 4
a.) Find the sum of the first 10 terms.
EXAMPLE 7: Consider
∞
P
b.) Estimate the error in using the sum of the first 10 terms as an approximation to
c.) Find
∞
P
n=1
n2
n=1 n2
1
.
+4
1
correct to 10 decimal places.
+4
EXAMPLE 8: How many terms of the series
1
?
81
∞
P
1
would you need to add to find its sum to within
n=1 n4
∞
P
3 + cos n
n5
n=1
a.) Prove the series converges.
EXAMPLE 10: Consider
b.) Approximate
∞
P
3 + cos n
using s6 .
n5
n=1
∞
P
c.) Estimate the error in using s6 to approximate
3 + cos n
.
n5
n=1
∞
P