354 STEP 4. Review the Knowledge You Need to Score High
14.4 Alternating Series
Main Concepts:Alternating Series, Error Bound, Absolute and Conditional Convergence
A series whose terms alternate between positive and negative is called an alternating series.
Alternating series have one of two forms:
∑∞
n= 1
(−1)nanor
∑∞
n= 1
(−1)n+^1 anwith allan’s>0. An
alternating series converges ifa 1 ≥a 2 ≥a 3 ≥···≥an≥···and limn→∞an=0.
Example 1
Determine whether the series
1
e
−
2
e^2
+
3
e^3
−
4
e^4
+···converges or diverges.
Step 1:
1
e
−
2
e^2
+
3
e^3
−
4
e^4
+···=
∑∞
n= 1
(−1)n+^1
n
en
Step 2: Note that
1
e
>
2
e^2
>
3
e^3
>
4
e^4
, and in general,
n
en
>
n+ 1
en+^1
, since multiplying by
en+^1 givesen>n+1.
Step 3:
{
1
e
,
2
e^2
,
3
e^3
,
4
e^4
,...
}
≈{.36788,.27067,.14936,.07326,...}, so limn→∞
n
en
=0.
Therefore, the series converges.
Example 2
Determine whether the series 4− 1 +
1
4
−
1
16
+···converges or diverges. If it converges,
find its sum.
Step 1: 4 − 1 +
1
4
−
1
16
+···is a geometric series witha=4 andr=
− 1
4
. Since|r|<1,
the series converges.
Step 2: S=
a
1 −r
=
4
1 −
− 1
4
=
4
5
4
=
16
5
= 3. 2
Error Bound
If an alternating series converges to the sumS, thenSlies between two consecutive partial
sums of the series. IfSis approximated by a partial sumsn, the absolute error|S−sn|is
less than the next term of the seriesan+ 1 , and the sign ofS−snis the same as the coefficient
ofan+ 1.
Example 1
4 − 1 +
1
4
−
1
16
+···converges to 3.2. This value is greater thansnfornodd, and less than
snforneven. IfSis approximated by the third partial sum,s 3 = 3 .25, the absolute error