486
8.4. ALTERNATING SERIES
8.4 Alternating Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is an alternating series?
• What does it mean for an alternating series to converge?
• Under what conditions does an alternating series converge? Why?
• How well does the nth partial sum of a convergent alternating series approximate
the actual sum of the series? Why?
• What is the difference between absolute convergence and conditional convergence?
Introduction
In our study of series so far, almost every series that we’ve considered has exclusively
nonnegative terms. Of course, it is possible to consider series that have some negative
terms. For instance, if we consider the geometric series
2 −
4
3
+
8
9
− · · · + 2
−2
3
n
+ · · · ,
which has a = 2 and r = −
2
3 , we see that not only does every other term alternate in sign,
but also that this series converges to
S =
a
1 − r
=
2
1 −
−
2
3
=
6
5
.
In Preview Activity 8.4 and our following discussion, we investigate the behavior of similar
series where consecutive terms have opposite signs.
Preview Activity 8.4. Preview Activity 8.3 showed how we can approximate the number
e with linear, quadratic, and other polynomial approximations. We use a similar approach
in this activity to obtain linear and quadratic approximations to ln(2). Along the way,
we encounter a type of series that is different than most of the ones we have seen so far.
Throughout this activity, let f (x) = ln(1 + x).
(a) Find the tangent line to f at x = 0 and use this linearization to approximate ln(2).
That is, find L(x), the tangent line approximation to f (x), and use the fact that
L(1) ≈ f (1) to estimate ln(2).
8.4. ALTERNATING SERIES
8.4 Alternating Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is an alternating series?
• What does it mean for an alternating series to converge?
• Under what conditions does an alternating series converge? Why?
• How well does the nth partial sum of a convergent alternating series approximate
the actual sum of the series? Why?
• What is the difference between absolute convergence and conditional convergence?
Introduction
In our study of series so far, almost every series that we’ve considered has exclusively
nonnegative terms. Of course, it is possible to consider series that have some negative
terms. For instance, if we consider the geometric series
2 −
4
3
+
8
9
− · · · + 2
−2
3
n
+ · · · ,
which has a = 2 and r = −
2
3 , we see that not only does every other term alternate in sign,
but also that this series converges to
S =
a
1 − r
=
2
1 −
−
2
3
=
6
5
.
In Preview Activity 8.4 and our following discussion, we investigate the behavior of similar
series where consecutive terms have opposite signs.
Preview Activity 8.4. Preview Activity 8.3 showed how we can approximate the number
e with linear, quadratic, and other polynomial approximations. We use a similar approach
in this activity to obtain linear and quadratic approximations to ln(2). Along the way,
we encounter a type of series that is different than most of the ones we have seen so far.
Throughout this activity, let f (x) = ln(1 + x).
(a) Find the tangent line to f at x = 0 and use this linearization to approximate ln(2).
That is, find L(x), the tangent line approximation to f (x), and use the fact that
L(1) ≈ f (1) to estimate ln(2).