CK-12-Calculus

8.4. Series With Odd or Even Negative Terms http://www.ck12.org

The Alternating Series Test

As its name implies, theAlternating Series Testis a test for convergence for series who have alternating signs in
its terms.
Theorem (The Alternating Series Test)
The alternating seriesu 1 −u 2 +u 3 −u 4 +...or−u 1 +−u 1 +u 2 −u 3 +u 4 −...converge if:
1.u 1 ≥u 2 ≥u 3 ≥...≥uk≥...
and

2. limk→+∞uk=0.
   Take the terms of the series and drop their signs. Then the theorem tells us that the terms of the series must be
   nonincreasing and the limit of the terms is 0 in order for the test to work. Here is an example of how to use The
   Alternating Series Test.
   Example 1
   Determine if∑∞k= 1 (− 1 )k+^1 kk 3 ++^5 kconverges or diverges.
   Solution
   The series∑∞k= 1 (− 1 )k+^1 kk 3 ++^5 kis an alternating series. We must first check that the terms of the series are nonincreas-
   ing. Note that in order foruk≥uk+ 1 , then 1≥uku+k^1 , oruku+k^1 ≤1.
   So we can check that the ratio of the(k+ 1 )st term to thekth term is less than or equal to one.

uk+ 1

uk =

(k+ 1 )+ 5

(k+ 1 )^3 +(k+ 1 )

kk^3 ++^5 k =

(k+ 1 )+ 5

(k+ 1 )^3 +(k+ 1 )×

k^3 +k

k+ 5

Expanding the last expression, we get:

uk+ 1

uk =

(k+ 6 )(k^3 +k)

(k^3 + 3 k^2 + 4 k+ 2 )(k+ 5 )=

k^4 + 6 k^3 +k^2 + 6 k

k^4 + 8 k^3 + 19 k^2 + 22 k+ 10.

Sincekis positive and all the sum of the numerator are part of the denominator’s sum, the numerator is less than the
denominator and so,uuk+k^1 <1. Thus,uk≥uk+1 for allk. By the Alternating Series Test, the series∑∞k= 1 (− 1 )k+^1 kk 3 ++^5 k
converges.
Keep in mind that both conditions have to be satisfied for the test to prove convergence. However, if the limit
condition is not satisfied, the infinite series diverges.

Alternating Series Remainder

We find the sequence of partial sums for an alternating series. A partial sum can be used to approximate the sum of
the series. If the alternating series converges, we can actually find a bound on the difference between the partial sum
and the actual sum. This difference, or remainder, is called theerror.
Theorem (Alternating Series Remainder)