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11.3 Alternating Series

11.29   Definition (Alternating series.) Series of the form or where for all are called alternating series.

11.30   Theorem (Alternating series test.) Let be a decreasing sequence of positive numbers such that . Then is summable. Moreover,

and

for all .

Proof: Let . For all ,

and

Thus is decreasing and is increasing. Also, for all ,

so is bounded below by , and

so is bounded above by .

It follows that there exist real numbers and such that

Now

so .

It follows from the next lemma that ; i.e.,

Since for all

we have

and since

Thus, in all cases, ; i.e., approximates with an error of no more than .

11.31   Lemma. Let be a real sequence and let . Suppose and . Then .

Proof: Let be a precision function for and let be a precision function for . For all , define

I claim is a precision function for , and hence . Let .
Case 1:
is even. Suppose is even. Say where . Then

Case 2:
is odd. Suppose is odd. Say where . Then

Hence, in all cases,

11.32   Remark. The alternating series test has obvious generalizations for series such as

and we will use these generalizations.

11.33   Example. If , then

are decreasing positive null sequences, so

are summable; i.e.,

(These are the sequences we called and in example 10.3.)

Also, , with an error smaller than
. My calculator says

and

11.34   Entertainment. Since is a decreasing positive null sequence for , it follows that converges for . We will now explicitly calculate the limit of this series using a few ideas that are not justified by results proved in this course. We know that for all , and all ,

Hence, for all ,

i.e.,

Thus

Hence

for all .

If we can show that is a null sequence, it follows that

or in other words,
 (11.35)

I claim is a null sequence for and hence (11.35) holds for . In particular,

First suppose , then for , so

Since is a null sequence for , it follows from the comparison test that is a null sequence for . Now suppose . Then

so and

If , then is a null sequence, so is a null sequence.

11.36   Entertainment. By starting with the formula

for all and using the ideas from the last example, show that
 (11.37)

Conclude that

11.38   Exercise. Determine whether or not the following series converge.
a)
b)
c)
(assume here ).

Next: 11.4 Absolute Convergence Up: 11. Infinite Series Previous: 11.2 Convergence Tests   Index