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8.3. SERIES OF REAL NUMBERS
Activity 8.11.
The series
1
k p are special series called p-series. We have already seen that the p-series
with p = 1 (the harmonic series) diverges. We investigate the behavior of other p-series
in this activity.
(a) Evaluate the improper integral
∞
1
1
x 2 dx. Does the series
∞
k=1
1
k 2 converge or
diverge? Explain.
(b) Evaluate the improper integral
∞
1
1
x p dx where p > 1. For which values of p
can we conclude that the series
∞
k=1
1
k p converges?
(c) Evaluate the improper integral
∞
1
1
x p dx where p < 1. What does this tell us
about the corresponding p-series
∞
k=1
1
k p ?
(d) Summarize your work in this activity by completing the following statement.
The p-series
∞
k=1
1
k p converges if and only if
.
⊳
The Limit Comparison Test
The Integral Test allows us to determine the convergence of an entire family of series: the
p-series. However, we have seen that it is, in general, difficult to integrate functions, so the
Integral Test is not one that we can use all of the time. In fact, even for a relatively simple
series like
k 2 +1
k 4 +2k+2
, the Integral Test is not an option. In this section we will develop a
test that we can use to apply to series of rational functions like this by comparing their
behavior to the behavior of p-series.
Activity 8.12.
Consider the series
k+1
k 3 +2
. Since the convergence or divergence of a series only
depends on the behavior of the series for large values of k, we might examine the terms
of this series more closely as k gets large.
(a) By computing the value of
k+1
k 3 +2
for k = 100 and k = 1000, explain why the
terms
k+1
k 3 +2
are essentially
k
k 3 when k is large.
(b) Let’s formalize our observations in (a) a bit more. Let a k =
k+1
k 3 +2
and b k =
k
k 3 .
Calculate
lim
k→∞
a k
b k
.
What does the value of the limit tell you about a k and b k for large values of k?
Compare your response from part (a).
(c) Does the series
k
k 3 converge or diverge? Why? What do you think that tells
us about the convergence or divergence of the series
k+1
k 3 +2
? Explain.
8.3. SERIES OF REAL NUMBERS
Activity 8.11.
The series
1
k p are special series called p-series. We have already seen that the p-series
with p = 1 (the harmonic series) diverges. We investigate the behavior of other p-series
in this activity.
(a) Evaluate the improper integral
∞
1
1
x 2 dx. Does the series
∞
k=1
1
k 2 converge or
diverge? Explain.
(b) Evaluate the improper integral
∞
1
1
x p dx where p > 1. For which values of p
can we conclude that the series
∞
k=1
1
k p converges?
(c) Evaluate the improper integral
∞
1
1
x p dx where p < 1. What does this tell us
about the corresponding p-series
∞
k=1
1
k p ?
(d) Summarize your work in this activity by completing the following statement.
The p-series
∞
k=1
1
k p converges if and only if
.
⊳
The Limit Comparison Test
The Integral Test allows us to determine the convergence of an entire family of series: the
p-series. However, we have seen that it is, in general, difficult to integrate functions, so the
Integral Test is not one that we can use all of the time. In fact, even for a relatively simple
series like
k 2 +1
k 4 +2k+2
, the Integral Test is not an option. In this section we will develop a
test that we can use to apply to series of rational functions like this by comparing their
behavior to the behavior of p-series.
Activity 8.12.
Consider the series
k+1
k 3 +2
. Since the convergence or divergence of a series only
depends on the behavior of the series for large values of k, we might examine the terms
of this series more closely as k gets large.
(a) By computing the value of
k+1
k 3 +2
for k = 100 and k = 1000, explain why the
terms
k+1
k 3 +2
are essentially
k
k 3 when k is large.
(b) Let’s formalize our observations in (a) a bit more. Let a k =
k+1
k 3 +2
and b k =
k
k 3 .
Calculate
lim
k→∞
a k
b k
.
What does the value of the limit tell you about a k and b k for large values of k?
Compare your response from part (a).
(c) Does the series
k
k 3 converge or diverge? Why? What do you think that tells
us about the convergence or divergence of the series
k+1
k 3 +2
? Explain.