358 STEP 4. Review the Knowledge You Need to Score High
14.6 Taylor Series
Main Concepts:Taylor Series and MacLaurin Series, Common MacLaurin SeriesTaylor Series and MacLaurin Series
A Taylor polynomial approximates the value of a functionf(x) at the pointx=a. If the
function and all its derivatives exist atx=a, then on the interval of convergence, the Taylor
series∑∞
n= 0f(n)(a)
n!
(x−a)nconverges to f(x). The MacLaurin series is the name given to a
Taylor series centered atx=0.
Example 1
Find the Taylor polynomial of degree 3 forf(x)=1
x+ 2
about the pointx=3.Step 1: Differentiate: f′(x)=− 1
(x+2)^2
, f′′(x)=2
(x+2)^3
,f′′′(x)=− 6
(x+2)^4.
Step 2: Evaluate:f(3)=1
5
, f′(3)=− 1
25
, f′′(3)=2
125
, f′′′(x)=− 6
625
.
Step 3:
f(3)
0!=
1 / 5
1
=
1
5
f′(3)
1!=
− 1 / 25
1
=
− 1
25
f′′(3)
2!=
2 / 125
2
=
1
125
f′′′(3)
3!=
6 / 625
6
=
1
125
Step 4:∑^3
n= 0f(n)(a)
n!
(x−a)n=1
5
−
(x−3)
25+
(x−3)^2
125−
(x−3)^3
625Example 2
A function f(x) is approximated by the third order Taylor series 1+ 2(x −1) −
(x−1)^2 +(x−1)^3 centered atx=1. Findf′(1) and f′′′(1).Step 1: Compare∑∞
n= 0f(n)(a)
n!
(x−a)nto the given polynomial:
f(1)
0!=1,
f′(1)
1!=2,
f′′(1)
2!=
−1, and
f′′′(1)
3!=1.
Step 2: f′(1)= 2 ·1!=2 andf′′′(1)= 1 ·3!=6.Example 3
Find the MacLaurin polynomial of degree 4 that approximatesf(x)=ln(1+x).Step 1: Differentiate: f′(x) =1
1 +x
, f′′(x)=− 1
(1+x)^2
, f′′′(x) =2
(1+x)^3,
f(4)(x)=− 6
(1+x)^4.
Step 2: Evaluate:f(0)=0,f′(0)=1,f′′(0)=−1,f′′′(0)=2,f(4)(0)=−6.