5 Steps to a 5 AP Calculus BC 2019

Series 365

For problems 6–8, determine whether each

series converges absolutely, converges

conditionally, or diverges.

6.

∑∞

n= 1

(−1)n−^1

n!

7.

∑∞

n= 1

(−1)n−^1

n+ 1

n

8.

∑∞

n= 1

(−1)n

n+ 1

7 n^2 − 5

9. Find the sum of the geometric series
   ∑∞
   n= 0

4

(

1

3

)n

.

10. If the sum of the alternating series
    ∑∞
    n= 1

(−1)n−^1

2 n− 1

is approximated bys 50 , find the

maximum absolute error.

Find the interval of convergence for each series.

11. 
    

∑∞

n= 0

xn

1 +n^2

12. ∑∞
    n= 1

3 n

n^2

xn

13. The Taylor series representation of lnx,
    centered atx=a.

Approximate each function with a fourth

degree Taylor polynomial centered at the given

value ofx.

14. f(x)=ex^2 atx= 1.


15. f(x)=cosπxatx=

1

2

.

16. f(x)=lnxatx=e.

Find the MacLaurin series for each function and

determine its interval of convergence.

17. f(x)=

1

1 −x

18. f(x)=

1

1 +x^2

19. Estimate sin 9◦accurate to three decimal
    places.


20. Find the rational number equivalent to
    1 .83.

14.10 Cumulative Review Problems

21. The movement of an object in the plane is
    defined byx(t)=lnt,y(t)=t^2. Find the
    speed of the object at the moment when
    the acceleration isa(t)=〈−1, 2〉.


22. Find the slope of the tangent line to the

curver=5 cos 3θwhenθ=

2 π

3

.

23.

∫e

1

x^3 lnxdx

24.

∫ 1

0

5

x^2 −x− 6

dx

25. limx→ 1
    lnx
    x^2 − 1

14.11 Solutions to Practice Problems

1.

∑∞

n= 0

5 −n=

∑ 1

5 n

= 1 +

1

5

+

1

25

+

1

125

+···is a geometric series with an initial

term of one and a ratio of

1

5

. Since the

ratio is less than one, the series converges,

and

∑∞

n= 0

5 −n=

1

1 − 1 / 5

=

5

4

.