5 Steps to a 5 AP Calculus BC 2019

364 STEP 4. Review the Knowledge You Need to Score High

∑∞

n= 1

1

√

n

, which is ap-series withp=

1

2

, and therefore diverges. Thus, the interval of

convergence is [−2, 0).

8. Approximate the functionf(x)=

1

x+ 2

with a fourth degree Taylor polynomial

centered atx=3.

Answer: f(3)=

1

5

,f′(x)=

− 1

(x+2)^2

⇒f′(3)=

− 1

25

,

f′′(x)=

2

(x+2)^3

⇒f′′(3)=

2

125

,f′′′(x)=

− 6

(x+2)^4

⇒ f′′′(3)=

− 6

625

,

f(4)(x)=

24

(x+2)^5

⇒ f(4)(3)=

24

3125

,so

P(x)=

1 / 5

0!

(x−3)^0 +

− 1 / 25

1!

(x−3)^1 +

2 / 125

2!

(x−3)^2

+

− 6 / 625

3!

(x−3)^3 +

24 / 3125

4!

(x−3)^4

=

1

5

−

x− 3

25

+

(x−3)^2

125

−

(x−3)^3

625

+

(x−3)^4

3125

.

9. Find the MacLaurin series for the functionf(x)=e−xand determine its interval of
   convergence.

Answer:Sinceex=

∑xn

n!

, substitute−xto find

e−x=

∑(−x)n

n!

= 1 −x+

x^2

2

−

x^3

6

+···. The ratio limn→∞

∣∣

∣

∣

(−x)n+^1

(n+1)!

n!

(−x)n

∣∣

∣

∣

=nlim→∞

∣∣

∣∣ −x

n+ 1

∣∣

∣∣=0, so the series converges on the interval (−∞,∞).

14.9 Practice Problems

For problems 1–5, determine whether each

series converges or diverges.

1.

∑∞

n= 0

5 −n

2.

∑∞

n= 1

1

n· 2 n

3.

∑∞

n= 0

n

en

4.

∑∞

n= 1

n+ 1

n(n+2)

5.

∑∞

n= 1

n

(n+1)n