354 STEP 4. Review the Knowledge You Need to Score High
14.4 Alternating Series
Main Concepts:Alternating Series, Error Bound, Absolute and Conditional ConvergenceA 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 series1
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
enStep 2: Note that1
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 thansnforneven. IfSis approximated by the third partial sum,s 3 = 3 .25, the absolute error