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CHAPTER 8 REVIEW 629

(c) If a series is convergent by the Alternating Series Test, how

do you estimate its sum?

8. (a) Write the general form of a power series.

(b) What is the radius of convergence of a power series?

(c) What is the interval of convergence of a power series?

9. Suppose is the sum of a power series with radius of con-

vergence .

(a) How do you differentiate ? What is the radius of conver-

gence of the series for ?

(b) How do you integrate ? What is the radius of convergence

of the series for ?

10. (a) Write an expression for the -degree Taylor polynomial

of centered at .af

nth

xfxdxf

f

f

R

fx

(b) Write an expression for the Taylor series of centered at .

(c) Write an expression for the Maclaurin series of .

(d) How do you show that is equal to the sum of itsTaylor series?

(e) State Taylors Inequality.

11. Write the Maclaurin series and the interval of convergence for

each of the following functions.

(a) (b)

(c) (d)

(e) (f)

12. Write the binomial series expansion of . What is the

radius of convergence of this series?

1 xk

ln1 xtan1x

cos xsin x

ex11 x

fx

f

af

11. If , then .

12. If is divergent, then is divergent.

13. If converges for all ,

then .

14. If and are divergent, then is divergent.

15. If and are divergent, then is divergent.

16. If is decreasing and for all , then is

convergent.

17. If and converges, then converges.

18. If and , then .

19.

20. If , then .limnl

an3 an 0limnl

an 2

0.99999 . . . 1

limnlan 0limnlan1an 1an 0

1nananan 0

an nan 0an

anbn bn an

an bn bn an

f0 2xfx 2x x 2

1

3x3

an an

limnln 01 1Determine whether the statement is true or false. If it is true, explain why.

If it is false, explain why or give an example that disproves the statement.

1. If , then is convergent.

2. The series is convergent.

3. If , then .

4. If is convergent, then is convergent.

5. If is convergent, then is convergent.

6. If diverges when , then it diverges when .

7. The Ratio Test can be used to determine whetherconverges.

8. The Ratio Test can be used to determine whether

converges.

9. If and diverges, then diverges.

10.

n0

1n

n!

1

e

anbn0 an bn

1n!

1n3

x 10x 6cnx ncn6ncn6ncn2ncn6n

limnla2n1 Llimnlan L

n1nsin 1anlimnlan 0

8. A sequence is defined recursively by the equations ,

. Show that is increasing and

for all . Deduce that is convergent and find its limit.

918 Determine whether the series is convergent or divergent.

9. 10.

11. 12.

13. 14.

n1

ln n3n 1

n2

1

nsln n

n1

1n

sn 1

n1

n3

5n

n1

n2 1

n3 1

n1

n

n3 1

an nan 2an an1

13 an 4

a1 117 Determine whether the sequence is convergent or divergent.

If it is convergent, find its limit.

1. 2.

3. 4.

5. 6.

7. 1 3n4n

an ln n

snan

n sin n

n2 1

an cosn2an n3

1 n2

an 9n1

10nan

2 n3

1 2n3

True-False Quiz

Exercises

; Graphing calculator or computer with graphing software required

8/2/2019 Exerccios - Sequncias e Sries

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630 CHAPTER 8 INFINITE SEQUENCES AND SERIES

15. 16.

17. 18.

1922 Find the sum of the series.

19. 20.

21.

22.

23. Express the repeating decimal as a

fraction.

24. For what values of does the series converge?

25. Find the sum of the series correct to four deci-

mal places.

26. (a) Find the partial sum of the series and estimate

the error in using it as an approximation to the sum of the

series.

(b) Find the sum of this series correct to five decimal places.

27. Use the sum of the first eight terms to approximate the sum ofthe series . Estimate the error involved in this

approximation.

28. (a) Show that the series is convergent.

(b) Deduce that .

29. Prove that if the series is absolutely convergent, then

the series

is also absolutely convergent.

3033 Find the radius of convergence and interval of convergence

of the series.

30. 31.

32. 33.

34. Find the radius of convergence of the series

n

1

2n!

n!2

x n

n0

2nx 3n

sn 3

n1

2nx 2n

n 2!

n1

x 2n

n 4n

n1

1nx n

n2 5n

n1

n 1n

ann1an

limnl

n n

2n! 0

n1

n n

2n!

n12 5n1

n1 1n6s5

n1

1n1

n 5

n1ln xnx

1.2345345345 . . .

1 e e 2

2!

e 3

3!

e 4

4!

n1

tan1n 1 tan1n

n0

1nn

32n2n!

n1

3n1

23n

n1

52n

n 2 9n

n1

1 3 5 2n 1

5nn!

n1

cos 3n

1 1.2n

n1

1n1sn

n 1

35. Find the Taylor series of at .

36. Find the Taylor series of at .

3744 Find the Maclaurin series for and its radius of conver-

gence. You may use either the direct method (definition of a

Maclaurin series) or known series such as geometric series,

binomial series, or the Maclaurin series for , , and .

37. 38.

39. 40.

41. 42.

43. 44.

45. Evaluate as an infinite series.

46. Use series to approximate correct to two deci-

mal places.

4748

(a) Approximate by a Taylor polynomial with degree at the

number .

; (b) Graph and on a common screen.(c) Use Taylors Inequality to estimate the accuracy of the approxi-

mation when lies in the given interval.; (d) Check your result in part (c) by graphing .

47. , , ,

48. , , ,

49. Use series to evaluate the following limit.

50. The force due to gravity on an object with mass at a

height above the surface of the earth is

where is the radius of the earth and is the acceleration due

to gravity.

(a) Express as a series in powers of .

; (b) Observe that if we approximate by the first term in theseries, we get the expression that is usually used

when is much smaller than . Use the Alternating Series

Estimation Theorem to estimate the range of values of for

which the approximation is accurate to within one

percent. (Use km.)R

6400

F mth

Rh

F mtF

hRF

tR

FmtR2

R h2

h

m

limxl0

sin x x

x 3

0 x 6n 2a 0fx sec x

0.9 x 1.1n 3a 1fx sxRnx

xfx Tnx

Tnf

a

nf

x10s1 x 4 dx

y ex

xdx

fx

1

3x5

fx

1s4

16x

fx 10xfx sinx 4

fx xe 2xfx ln4 x

fx tan1x 2 fx x 2

1 x

tan1xsin xex

f

a

3fx

cos x

a 6fx sin x