February Thursday, 10 Lecture 20 : Alternating series (Refers to Section 8.2 in
your text)
After having practiced the problems associated to the concepts of this lecture the student should
be able to: Define an alternating series, state the alternating series test, apply the
alternating series test.
20.1 Definition An alternating series is a series of the form
where cj is non-negative for all j. (In practice, the series need not start with j = 1. It can
also start with j equal to any positive integer.). Also the series need not necessarily start
with a positive term.
One normally calls on the Alternating series test to test such series for convergence. It is
described and proven below :
20.2 Theorem The alternating series test. (AST) If {cj} is a sequence of non-negative
numbers such that 1) limj cj = 0 and 2) the terms of {cj} are strictly decreasing, then
the alternating series
converges.
Proof:
Suppose limj cj = 0 and {cj} is strictly decreasing.
Let {Sn : n = 1 to } be the partial sums of the series.
First observe that
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The sequence {S2n : n = 1 to } is a monotone increasing sequence of positive
terms and that it is bounded above by the positive number c1.
-
So by the Monotone sequence theorem, the increasing sequence {S2n : n = 1 to }
converges to some number S.
Next note that the sequence {S2n + 1 : n = 0 to } is a decreasing sequence and that
S2n +1 = S2n + c2n + 1. Hence
-
Thus the sequence {S2n + 1 : n = 0 to } converges to S.
So both {S2n : n = 1 to } and {S2n + 1 : n = 0 to } converge to S.
We claim that {Sn : n = 1 to } must also converge to S :
Since the partial sums {Sn : n = 1 to } converge to S then the alternating series
converges to S.
20.3 Remark It will be useful to remember that the even terms {S2n : n = 1 to } of
partial sums are increasing towards S while the odd terms {S2n + 1 : n = 0 to } of partial
sums are monotone decreasing towards S.
20.3.1 Definition The series
is called the alternating harmonic series.
20.3.2 Example We show that the alternating harmonic series converges.
For the alternating harmonic series we see that:
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The terms of {1/j : j = 1, 2, 3, ...} are all positive,
-
The sequence is decreasing
-
Lim j cj = 0.
By the alternating series test the series converges.
20.4 Example Does the series
converge or diverge?
We will begin by testing for divergence the using Divergence test. We verify in class
that the terms go to zero and so the Divergence test fails.
-
We apply the Alternate series test.
-
We must check that conditions are satisfied:
This is indeed an alternating series (follows from the definition.)
When we did the Divergence test we already saw that terms go to zero.
To verify that the (positive part of the terms) is strictly decreasing, we check
that the ratio cn+1/cn is always less than 1:
20.5 Example Does the following series converge or diverge?
Solution:
-
We apply the Divergence test: We easily see that
So Divergence test fails
-
We then apply the alternating series test:
a) We show that {(ln n )/n} is decreasing (by taking the derivative of (ln x ) / x)
b) We already saw above that limn (ln n )/n = 0.
By the Alternating series test our original series converges conditionally.
20.6 Example Does the following series converge or diverge?
Solution outline:
We apply the Divergence test: We easily see that
So Divergence test fails.
-
We test for convergence using the Alternate series.
-
We must check that conditions are satisfied:
This is indeed an alternating series (follows from the definition.)
When we did the Divergence test we already saw that terms go to zero.
To verify that they are strictly decreasing:
By the alternating series test our series converges.
20.7 Remark We have seen various techniques for determining whether a sequence is
monotone decreasing or monotone increasing. We summarize these here for the case
where we want to show monotone increasing. Given a series {an : n = 1, 2, 3, .}.
1. In simple cases show directly that an+1 > an.
- For example, for {1/n2 : 1, 2, 3, .}, 1/(n + 1)2 > 1/n2 for all n and so the sequence
is increasing. Also, example 20.6 above illustrates this for the case monotone
decreasing.
2. Show that an+1/an > 1 for all n as we did in example 20.4 above.
3. If f (n) = an and f is continuous on [1, ) verify if f (x) > 0 on [1, ) eg.: {n/(ln n )}
4. In cases where the sequence is recursive, use the Principle of mathematical induction.
- At first students may not feel comfortable applying this principle. But like many
things, by using it in various situations they develop a better understanding of the
principle and eventually see why it is sometimes the only way to show that a
statement is true.