Series 359
Step 3:
f(0)
0!
x^0 +
f′(0)
1!
x^1 +
f′′(0)
2!
x^2 +
f′′′(0)
3!
x^3 +
f(4)(0)
4!
x^4 =
0
1
x^0 +
1
1
x^1 +
− 1
2
x^2 +
2
6
x^3 +
− 6
24
x^4 =x−
1
2
x^2 +
1
3
x^3 −
1
4
x^4
Example 4
Find the Taylor series for the functionf(x)=e−xabout the pointx=ln 2.
Step 1: f(n)(x)=e−xwhennis even andf(n)(x)=−e−xwhennis odd.
Step 2: Evaluatef(n)(ln 2)=e−ln 2=
1
2
whennis even andf(n)(ln 2)=
− 1
2
whennis odd.
Step 3: f(x)=e−x =
1 / 2
0!
(x − ln 2)^0 +
− 1 / 2
1!
(x − ln 2)^1 +
1 / 2
2!
(x −ln 2)^2 + ···
=
∑∞
n= 0
(−1)n
2 ·n!
(x −ln 2)n
Example 5
Find the MacLaurin series for the functionf(x)=xex.
Step 1: Investigating the first few derivatives of f(x)=xexshows thatf(n)(x)=xex+nex.
Step 2: Evaluating f(n)(x)=xex+nexatx=0 givesf(n)(0)=n.
Step 3: f(x)=
∑∞
n= 0
f(n)(0)
n!
xn=
∑∞
n= 0
n
n!
xn=
∑∞
n= 1
xn
(n−1)!
Common MacLaurin Series
MacLaurin Series for the Functionsex,sinx,cosx,and 11 −x
Familiarity with these common MacLaurin series will simplify many problems.
f(x)=ex=
∑∞
n= 0
xn
n!
= 1 +x+
x^2
2
+
x^3
6
+···
f(x)=sinx=
∑∞
n= 0
(−1)nx^2 n+^1
(2n+1)!
=x−
x^3
3!
+
x^5
5!
−
x^7
7!
+···
f(x)=cosx=
∑∞
n= 0
(−1)^2 nx^2 n
(2n)!
= 1 −
x^2
2
+
x^4
24
−
x^6
6!
+···
f(x)=
1
1 −x
=
∑∞
n= 0
xn= 1 +x+x^2 +x^3 +···
14.7 Operations on Series
Main Concepts:Substitution, Differentiation and Integration, Error Bounds
Substitution
New series can be generated by making an appropriate substitution in a known series.