5 Steps to a 5 AP Calculus BC 2019

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.