360 STEP 4. Review the Knowledge You Need to Score High
Example 1
Find the MacLaurin series forf(x)=
1
1 +x^2
.
Step 1: Begin with the known series f(x)=
1
1 −x
=
∑∞
n= 0
xn.
Step 2: Substitute−x^2 forx.
1
1 +x^2
=
∑∞
n= 0
(−x^2 )n=
∑∞
n= 0
(−1)nx^2 n= 1 −x^2 +x^4 −x^6 +···
Example 2
Find the first four non-zero terms of the MacLaurin series forf(x)=cos(2x).
Step 1: Begin with the known series cosx= 1 −
x^2
2!
+
x^4
4!
−
x^6
6!
+···
Step 2: Substitute 2xforx. cos(2x)= 1 −
(2x)^2
2!
+
(2x)^4
4!
−
(2x)^6
6!
+···= 1 −
4 x^2
2
+
16 x^4
24
−
64 x^6
720
+···= 1 − 2 x^2 +
2
3
x^4 −
4
45
x^6
Differentiation and Integration
If a functionf(x) is represented by a Taylor series with a non-zero radius of convergence,
the derivative f′(x) can be found by differentiating the series term by term. If the series
is integrated term-by-term, the resulting series converges to
∫
f(x)dx. In either case, the
radius of convergence is identical to that of the original series.
Example 1
Differentiate the MacLaurin series forf(x)=ln(x+1) to find the Taylor series expansion
forf(x)=
1
x+ 1
.
Step 1: f(x)=ln(x+1)=x−
1
2
x^2 +
1
3
x^3 −
1
4
x^4 +···
Step 2: f′(x)=
1
x+ 1
= 1 −x+x^2 −x^3 +···=
∑∞
n= 0
(−1)nxn
Example 2
Find the MacLaurin series forf(x)=
1
(x+1)^2
.
Step 1: We know that
1
x+ 1
= 1 −x+x^2 −x^3 +···=
∑∞
n= 0
(−1)nxn.