right triangle ABC is given with theta= 1.1 and |AC| = b = 4. CD is drawnperpendicular to AB, DE is drawn perpendicular to BC, EF AB and thisprocess is continued indefinitely as shown in the figure. Find the total length ofall the perpendiculars |CD| + |DE| + |EF| + |FG| + ... 

The questions in this document with figure. How to solve this. 

 

 

Stewart - Calculus 6e Chapter 12 Form A 1. Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms
continues.
⎧ 1 1 1 1
⎫
, ,... ⎬
⎨1, , ,
3
9
27
81
⎩
⎭ 2. Use the Integral Test to determine whether the series is convergent or divergent.
∞ ∑
n =1 1
8n + 1 3. Determine whether the sequence converges or diverges. If it converges, find the limit.
an = e1/ n 4. Determine whether the sequence converges or diverges. If it converges, find the limit. an = n 32 n +1
5. Determine whether the sequence defined as follows is convergent or divergent.
a1 = 1 , an +1 = 4 − an for n ≥ 1 6. Test the series for convergence or divergence.
∞ ∑
n =5 7. Test the series for convergence or divergence.
∞ ∑
k =1 8. n 2 − 25
n 2 + 5n (−3) k +1
4 2k Find the values of p for which the series is convergent.
∞ ∑ n(1 + n 2 p ) n =1 9. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
an = 1
2n + 3 10. Test the series for convergence or divergence.
∞ ∑ m =1 ( m 3 − 1) m Stewart - Calculus 6e Chapter 12 Form A 11. Determine whether the series is conditionally convergent, absolutely convergent, or divergent.
∞ ∑ (−1) n −1 n −1/ 3 n =1 12. Find the interval of convergence of the series.
∞ ∑
n =1 6xn
5 n 13. Find the radius of convergence of the series.
∞ ∑
n =1 8n x n
(n + 5) 2 14. Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
∞ 1 ∑ 1+ 2
n =1 n 15. Determine the interval of convergence for the function.
⎛ y⎞
f ( y ) = arctan⎜ ⎟
⎝9⎠ 16. Find the sum of the series.
∞ ∑ n(n − 1) x n +1 ,| x | <1 n =2 17. Find the Maclaurin series for f and its radius of convergence.
f ( x) = ln(1 − x ) 18. Evaluate the indefinite integral as an infinite series. ∫ sin(4 x) dx
2 19. Find the Taylor polynomial T2 for the function f at the number a = 1.
f ( x) = x 20. Use the binomial series to expand the function as a power series. Find the radius of convergence.
1
(2 + x) 2 ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form A 1
3n −1 1. an = 2. divergent 3. 1 4. 9 5. divergent 6. divergent 7. convergent 8.
9. p < −1 Decreasing, yes 10. convergent
11. CC
12. [-1, 1)
13. 1/8
14. 0.76352, error < 0.001
15. (-9, 9)
16. 2 x3
(1 − x)3 17. −∑ ∞ n=0 xn
, R =1
n
∞ 18. C+ ∑
n =0 19. T2 = 1 + 20. |x| <2 (−1) n 4 4 n + 2 x 4 n +3
(4n + 3)(2n + 1)! 1
1
( x − 1) − ( x − 1) 2
8
2 Stewart - Calculus 6e Chapter 12 Form B 1. Find a formula for the general term an of the sequence, assuming that the pattern of the first few
terms continues.
1 1
1
⎧
⎫
⎨1, − , ,− , . . .⎬
3
9
27
⎩
⎭ 2. Find the value of the limit for the sequence.
⎧
⎛ 3n ⎞⎫
⎟⎬
⎨arctan⎜
⎝ 3n + 8 ⎠⎭
⎩ 3. Find the exact value of the limit of the sequence defined by a1 = 3 , an+1 = 3 + an . 4. Use the Integral Test to determine whether the series is convergent or divergent.
∞ ∑
n =1 1
3n + 1
∞ 5. Find the partial sum s10 of the series ∑
n =1 ∞ 6. Find the partial sum s7 of the series ∑ m =1 ∞ 7. How many terms of the series ∑ m =2 9
. Approximate the answer to the nearest thousandth.
n5 3
. Give your answer to five decimal places.
10 + 7 m 18
would you need to add to find its sum to within 0.02?
9m(ln m) 2 ∞ 8. Find all positive values of b for which the series ∑ 4b ln 3n converges. n =1 9. Write the partial sum of the converging series which represent the decimal number 0.2523. 10. Test the series for convergence or divergence.
∞ ∑
n=2 (−1) n n
5 ln n 11. Approximate the sum to the indicated accuracy.
∞ ∑
n =1 (−1) n −1
n7 (five decimal places) Stewart - Calculus 6e Chapter 12 Form B 12. Approximate the sum to the indicated accuracy.
∞ ∑
n=0 (−1) n
3n n! (four decimal places) 13. Test the series for convergence or divergence.
∞ ∑
k =5 5
k (ln k ) 6 14. Test the series for convergence or divergence.
∞ ∑
m=1 3m m 5
m! 15. Test the series for convergence or divergence.
∞ ∑ cos k k =1 16. Test the series for convergence or divergence.
∞ ∑ (−1) m ln m m =1 m 17. Test the series for convergence or divergence.
∞ ∑
k =1 k ln k
(k + 4) 3 18. Use Taylor's Inequality to estimate the accuracy |R4| of the approximation f (x) at the number a = 1,
when 0 ≤ x ≤ 2 .
f ( x) = cos( x) 19. A car is moving with speed 23 m/s and acceleration 2m/s2 at a given instant. Using a second-degree
Taylor polynomial, estimate how far the car moves in the next second.
20. The resistivity of a conducting wire is the reciprocal of the conductivity and is measured in units of
ohm-meters. The resistivity of a given metal depends on the temperature according to the equation
ρ (t ) = ρ 27 eα (t − 27 ) where t is the temperature in C. There are tables that list the values of α (called the temperature coefficient) and ρ 27 (the resistivity at 27 D C) for various metals. Except at very low
temperatures, the resistivity varies almost linearly with temperature and so it is common to
approximate the expression for ρ (t ) by its first-degree Taylor polynomial at t = 27. Find the expression
for this linear approximation. ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form B 1.
2. ⎛ 1⎞
an = ⎜ − ⎟
⎝ 3⎠ n −1 π
4 3. 1 + 13
2 4. divergent 5. 9.332 6. 0.23727 7. m > e100 8. b< 9. 1
5
2
3
+
+
+
5 10 2 103 10 4 1
e 10. divergent
11. 0.99259
12. 0.7165
13. convergent
14. convergent
15. divergent
16. convergent
17. convergent
18. 0.0083
19. 24
20. ρ 27 (1 + α (t − 27)) Stewart - Calculus 6e Chapter 12 Form C 1. Determine whether the sequence converges or diverges.
2n a ( n) = 5 n +1 Select the correct answer.
converges a.
2. b. diverges Find the value of the limit for the sequence given.
⎧⎪ 1 ⋅ 9 ⋅ 17 ⋅ ⋅ ⋅ (8n + 1)
⎨
(8n) 2
⎪⎩ ⎫⎪
⎬
⎪⎭ Select the correct answer.
a.
3. π b. -1 c. 3 d. 0 e. 1 A sequence {an } is defined recursively by the equation an = 0.5(an −1 + an − 2 ) for n ≥ 3 where
a1 = 18, a2 = 9 . Use your calculator to guess the limit of the sequence.
Select the correct answer.
a. 12 b. 13 c. 6 d. 17
∞ 4. Find the partial sum s7 of the series ∑
n =1 e. 26 1
. Give your answer to five decimal places.
2 + 5n Select the correct answer.
a. s7 = 0.18976 b. s7 = 0.18985
∞ 5. ∑ Given the series m =1 ∞ series ∑ m =9 c. s7 = 1.60976 d. s7 = 0.18975 e. s7 = 0.19176 3m
estimate the error in using the partial sum s8 by comparison with the
4 (3m + 5)
m 1
.
4m Select the correct answer.
a. R8 ≥ 0.0000051 b. R8 ≤ 2.6130051 c. R8 ≤ 0.0000051 d. R8 ≤ 0.000005 e. R8 ≥ 0.0000052 ∞ 6. Find all positive values of u for which the series ∑ 4u ln 7 m converges. m =1 Select the correct answer.
a. u < 4 b. 4 < u < 7
e c. u > 7 d. 0 < u < 1
e e. u > ln7 Stewart - Calculus 6e Chapter 12 Form C 7. Evaluate the function f ( x) = cos x by a Taylor polynomial of degree 4 centered at a = 0, and x = 2.
Select the correct answer.
a. 0.67 8. b. -0.33 c. -1.33 d. 0.28 e. -0.44 Given the series :
∞ A= ∑
k =1 1
and B =
5
k +9 ∞ ∑
k =1 1
k −k
4 Select the correct answer.
a.
b.
c.
d.
e.
9. Both series are convergent.
Both series are divergent.
Series A diverges by the Integral Test.
Series B and A converges by the Limit Comparison Test.
Series B diverges by the Integral Test. Test the series for convergence or divergence.
8
8
8
8
8
−
+
−
+
− ⋅⋅⋅
ln 2 ln 3 ln 4 ln 5 ln 6 Select the correct answer.
a. the series is divergent
b. the series is convergent
10. Use the power series for f ( x) = 3 5 + x to estimate 3 5.08 correct to four decimal places. Select the correct answer.
a. 1.7156 b. 1.7189 c. 1.7195 d. 1.7200 e. 1.7190 11. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?
∞ ∑
n =1 (−1) n +1
, (| error |) < 0.0399
n2 Select the correct answer.
a. n = 12 b. n = 5 c. n = 13 d. n = 6 12. Which of the given series are absolutely convergent? Select the correct answer.
∞ a. ∑
n =1 sin 3n
n ∞ b. ∑
n =1 cos πn 8
n n e. n = 8 Stewart - Calculus 6e Chapter 12 Form C 13. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞ ∑
n =1 (−1) n arctan n
n5 Select the correct answer.
a. divergent b. absolutely convergent c. conditionally convergent 14. Test the series for convergence or divergence.
∞ ∑ m =1 (−6) m +1
45m Select the correct answer.
a. The series is convergent b. The series is divergent 15. Test the series for convergence or divergence.
∞ ∑ (m 5 − 1) m m =1 Select the correct answer.
a. The series is divergent b. The series is convergent 16. Find the interval of convergence of the series.
∞ ∑
n =1 (−1) n x n
n+4 Select the correct answer.
a. [-1, 1) b. [-1, 1] c. (-1, 1] d. (-1, 1) e. diverges everywhere ∞ 17. Suppose that the radius of convergence of the power series ∑
n =0 ∞ convergence of the power series ∑
n =0 Select the correct answer.
a.
b.
c.
d.
e. 256
4
1
16
252 cn x 2 n ? cn x n is 16. What is the radius of Stewart - Calculus 6e Chapter 12 Form C 18. Evaluate the indefinite integral as a power series. ∫ tan −1 (t 2 ) dt Select the correct answer.
∞ a. C+ ∑
n=0 ∞ b. C+ ∑
n=0 ∞ c. C+ ∑
n=0 ∞ d. C+
C+ (−1) n t 4 n + 3
(4n + 3)
(−1) n t 4 n + 2
(2n + 1)(4n + 3) ∑ (−1) n t 2 n + 2
(2n + 1) ∞ (−1) n t 2 n + 3
(2n + 3) n=0 e. (−1) n t 4 n + 3
(2n + 1)(4n + 3) ∑
n=0 19. Use a power series to approximate the definite integral.
0.4 ∫ z 4 tan −1 ( z 4 ) dz 0 Select the correct answer.
a. 0.4 b. 0.000029 c. 1.400262 d. 1.399738 e. 1.004262 20. Find the Maclaurin series for f(x) using the definition of the Maclaurin series.
f ( x) = x cos(4 x) Select the correct answer.
∞ a. ∑ (−1) n 4 2 n x 2 n +1
n! ∞ (−1) n 4 2 n x 2 n +1
(2n)! n=0 b. ∑
n=0 ∞ c. ∑
n =0
∞ d. ∑
n=0
∞ e. ∑
n=0 (−1) n 4 2 n x 2 n
(2n)!
(−1) n +1 4 2n x 2 n +1
(2n)!
(−1) n 4 n x 2 n +1
(2n)! ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form C 1. a 2. d 3. a 4. a 5. c 6. d 7. b 8. a 9. b 10. e
11. b
12. b
13. b
14. a
15. b
16. c
17. b
18. a
19. b
20. b Stewart - Calculus 6e Chapter 12 Form D 1. Find the value of the limit of the sequence defined by a1 = 3 , a n +1 = 3 + a n .
Select the correct answer.
a. 2. 1 − 13
2 b. 1 + 13
2 c. 3
2 d. −1
2 e. 5
2 Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
an = 1
2n + 1 Select the correct answer.
a. decreasing
3. b. not monotonic c. increasing d. bounded Find the value of the limit of the sequence defined by a1 = 1, an +1 = 3 − 1
.
an Select the correct answer.
a.
4. 3+ 5
2 b. 3− 5
2 c. 3 − 10
2 c. − 77 + 7
14 d. 3 + 10
2 e. 3 + 10 Find the value of c.
∞ ∑ (1 + c) −n =7 n=2 Select the correct answer.
a. 0 5. b. 77 − 7
14 Given the two series A = 1 + d. − 1 1 1
1
+ +
+
+ ... and B =
4 9 16 25 ∞ ∑
n =1 is convergent or divergent and choose the correct statement.
Select the correct answer.
a.
b.
c.
d. Both series are divergent.
Series A is convergent, series B is divergent.
Both series are convergent.
Series A is divergent, series B is convergent. 7 +7
7 e. 7 −7
2 n 5e − n determine whether each series
6 Stewart - Calculus 6e Chapter 12 Form D ∞ 6. Find the partial sum s7 of the series ∑
k =1 3
. Give your answer to five decimal places.
4 + 5k Select the correct answer.
a. s7 = 0.46301
7. b. s7 = 0.47999 c. s7 = 2.276 d. s7 = 0.466 e. s7 = 0.566 Given the series:
∞ A= ∑ ∞ sin 2 5m m =1 10 m and B = m ∑ m =1 ⎛ 1 ⎞
8 cos⎜
⎟
⎝ 7m ⎠ Determine whether each series is convergent or divergent.
a.
b.
c.
d.
8. A is convergent, B is divergent.
A is divergent, B is convergent.
Both series are convergent.
Both series are divergent. Find the interval of convergence of the series.
∞ ∑
n =1 xn
n8n Select the correct answer.
a. (-8, 8]
9. b. (-1,1) c. [-1, 1] d. [-8, 8] e. [-8, 8) f. diverges everywhere Find the radius of convergence of the series.
∞ ∑
n =1 n3 x n
3n Select the correct answer.
a. R = ∞ b. R = 0 c. R = 1
3 d. R = 1 e. R = 3 10. Find the interval of convergence of the series.
∞ ∑ (−1) n n =1 ( x + 8) n
n6 n Select the correct answer.
a. (2, 14] b. (-14, -2) c. [-1,1] d. (-8, 6] e. [-14, -2) f. diverges everywhere Stewart - Calculus 6e Chapter 12 Form D 11. Find a power series representation for
f (t ) = ln(10 − t ) Select the correct answer.
∞ a. ∑ tn
n10 n ∞ 10t n
nn n =0 b. ∑
n =1 ∞ c. ln 10 − tn
10 n ∑
n =1
∞ d. ln 10 − tn
n10 n ∑
n =1
∞ e. ln 10 + t 2n
10 n ∑
n =1 12. Find a power series representation for the function.
⎛7+
f ( y ) = ln⎜⎜
⎝7− y⎞
⎟
y ⎟⎠ Select the correct answer.
∞ a. ∑ 14 y 2 n +1 ∞ y 2 n +1
7 n=0 b. ∑
n=0
∞ c. ∑
n=7 14 y 2 n +1
14 ∞ d. 2 y 2 n +1
7 n +1 (2n + 1) ∑
n=0
∞ e. ∑
n=0 y 2 n +1
7 n +1 (n + 1) 13. Use series to approximate the definite integral to within the indicated accuracy.
0.9 ∫xe 2 − x2 dx , | error | < 0.001 0 Select the correct answer.
a. 0.1249 b. 0.0125 c. 0.1449 d. 0.0625 e. 0.0825 Stewart - Calculus 6e Chapter 12 Form D 14. Use series to evaluate the limit correct to three decimal places.
7 x − tan −1 7 x
x →0
x3
lim Select the correct answer.
a. 114.133 b. 114.333 c. 34.3233 d. 115.933 e. 118.933 15. Use the binomial series to expand the function as a power series. Find the radius of convergence.
1
(1 + x) 4 Select the correct answer.
a. | x | < 8 b. | x | < 10 c. | x | < 1 d. | x | < 0.1 e. | x | < 1.8 16. Use the binomial series to expand the function as a power series. Find the radius of convergence. x
16 + x 2
Select the correct answer.
a. | x | < 8 b. | x | < 1 c. | x | < 4 d. | x | < 10 e. | x | < 1.8 17. Find the Taylor polynomial T3 for the function f at the number a = 1.
f ( x) = ln x Select the correct answer.
1
1
( x − 1) 2 + ( x − 1) 3
3
2
1
1
( x − 1) + ( x − 1) 2 + ( x − 1) 3
4
5
1
1
2
( x + 1) − ( x + 1) + ( x + 1) 3
3
4
1
1
2
( x + 1) − ( x + 1) + ( x + 1) 3
7
5
1
1
2
( x − 1) − ( x − 1) − ( x − 1) 3
7
5 a. ( x − 1) −
b.
c.
d.
e. Stewart - Calculus 6e Chapter 12 Form D 18. Find the Taylor polynomial T3 for the function f ( x) at the number a = 9.
f ( x) = cos x Select the correct answer.
sin(9)
cos(9)
( x + 9) 3
( x + 9) 2 +
6
2
sin(9)
cos(9)
( x − 9) 3
( x − 9) 2 +
cos(9) − sin(9)( x − 9) −
6
2
sin(9)
cos(9)
( x − 9) 3
( x − 9) 2 +
cos(9) + sin(9)( x − 9) +
6
2
sin(6)
cos(6)
( x − 9) 3
( x − 9) 2 +
cos(6) + sin(6)( x − 9) +
9
6
sin(2)
cos(2)
( x − 9)3
( x − 9) 2 +
cos(6) + sin(6)( x − 9) +
6
2 a. cos(9) − sin(9)( x + 9) −
b.
c.
d.
e. 19. Use Taylor's Inequality to estimate the accuracy |R2| of the approximation f ( x) at the number a = 1, when 0.71 ≤ x ≤ 1.29 .
f ( x) = 1
x2 Select the correct answer.
a.
b.
c.
d.
e. 1.0407
0.5407
1.5407
1.7407
1.8407 D
20. Estimate sin(35 ) correct to five decimal places. Select the correct answer.
a.
b.
c.
d.
e. 1.07358
1.57358
0.57358
1.77358
2.57358 ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form D 1. b 2. a, d 3. a 4. b 5. c 6. d 7. a 8. e 9. e 10. b
11. d
12. d
13. a
14. b
15. c
16. c
17. a
18. b
19. b
20. c Stewart - Calculus 6e Chapter 12 Form E 1. Determine whether the sequence converges or diverges.
an = 5 + 5n 2
n + n2 Select the correct answer.
a.
b. converges
diverges 2. If $1,600 is invested at 7% interest, compounded annually, then after n years the investment is worth
an = 1,600(1.07) n dollars. Find the size of investment after 7 years. 3. Determine whether the series is convergent or divergent. If it is convergent, write its sum. Otherwise
write divergent.
∞ ∑
n =1 ⎛3⎞
7⎜ ⎟
⎝4⎠ n 1 4. Express the number 0.87 as a ratio of integers. 5. Use the binomial series to expand the function as a power series. Find the radius of convergence.
4 6. 1 + x6 When money is spent on goods and services, those that receive the money also spend some of it. The
people receiving some of the twice-spent money will spend some of that, and so on. Economists call
this chain reaction the multiplier effect. In a hypothetical isolated community, the local government
begins the process by spending D dollars. Suppose that each recipient of spent money spends 100c%
and saves 100 s % of the money that he or she receives. The values c and s are called the marginal
propensity to consume and the marginal propensity to save and, of course, c + s = 1.
The number k = 1/s is called the multiplier. What is the multiplier if the marginal propensity to
consume is 80%?
Select the correct answer.
a. 6 b. 4 c. 5 d.
∞ 7. Find the partial sum s7 of the series ∑ m =1 7 5
.
6 + 9m Write your answer to five decimal places. e. 3 Stewart - Calculus 6e Chapter 12 Form E 8. Use the sum of the first 9 terms to approximate the sum of the following series:
∞ ∑ 6
n6 + n2 n =1 Write your answer to six decimal places.
9. Determine whether the series converges or diverges.
∞ 4 + cos n
4n ∑
n =1 Select the correct answer.
neither
converges
diverges a.
b.
c. ∞ 10. For what values of k does the series ∑
n =3 1
converge?
n k ln n 11. Test the series for convergence or divergence.
∞ ∑
n=2 (− 1)n −1
3 7 ln n 12. Find the radius of convergence of the series.
∞ ∑
n =1 5n x n
(n + 8) 3 13. Find the Maclaurin series for f ( x) using the definition of a Maclaurin series.
f ( x) = (2 + x) −3 14. Find the Taylor series for f ( x) centered at a = 1.
f ( x) = 2 + x + x 2 15. Use the binomial series to expand the function as a power series. Find the radius of convergence.
x2
2+ x Stewart - Calculus 6e Chapter 12 Form E 16. Find the Maclaurin series for f (x ) .
f ( x) = x cos(6 x) 17. Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin
series for the function.
f ( x) = e − x cos 4 x
2 Select the correct answer.
1 − 17 x 2 + 19.17 x 4 a.
b.
c.
d.
e. 1 − 9 x 2 + 11.17 x 4
1 − 9 x 2 + 19.17 x 4
1 − 9 x + 19.17 x 2
1 − 17 x 2 + 11.17 x 4 18. Use the binomial series to expand the function as a power series. Find the radius of convergence.
1
(5 + x) 6 19. The following table contains the evaluation of the Taylor polynomial centered at a = 1 for f ( x) = 1 / x . What is the degree of this polynomial?
x T(x) 0.5 1.88 0.7 1.42 1.7 0.45 2.8 -3.39
3 -5.00 Select the correct answer.
a. 3 b. 2 c. 1 d. 5 e. 4 20. Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values
of x for which the given approximation is accurate to within the stated error. cos x ≈ 1 − x2 x4
+
, | error | < 0.08
2 24 Write a such that −a < x < a . ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form E 1. a 2. 2569.25 3. 28 4. 29/33 5. |x| < 1 6. c 7. 0.39846 8. 3.098422 9. b 10. k >1 11. convergent
12. 1/5 13. ∞ ∑
n=0 ⎛x⎞
(−1) n (n + 1)(n + 2)⎜ ⎟
⎝2⎠
4
2 14. 4 + 3( x − 1) + ( x + 1) 2 15. |x| < 2
∞ 16. ∑ (−1) n 6 2 n x 2 n +1 n=0 (2n) ! 17. c
18. |x| < 5 19. a
20. −1.965 < x < 1.965 n Stewart - Calculus 6e Chapter 12 Form F 1. Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms
continues. { 1, 7, 13, 19, . . . }
2. Find the value of the limit for the sequence.
⎧⎪ 1 ⋅ 8 ⋅ 15 ⋅ ⋅ ⋅ (7 n + 1)
⎨
⎪⎩
(7 n) 2 3. ⎫⎪
⎬
⎪⎭ If $600 is invested at 4% interest, compounded annually, then after n years the investment is worth
an = 600(1.04) n dollars. Find the size of investment after 7 years.
Select the correct answer.
a. $789.56 4. b. $430.21 c. $1,321.06 e. $1,860.81 d. $1,230.81 Find the value of the limit of the sequence defined by a1 = 1, an +1 = 3 − 1
.
an Select the correct answer.
a.
5. 3 + 10
2 b. 3 − 10
2 c. 3− 5
2 d. 3+ 5
2 e. 2+ 5
3 Use Taylor's Inequality to estimate the accuracy |R3| of the approximation f (x) at the number a = 0,
when 0 ≤ x ≤ 0.5 .
f ( x) = tan x 6. A right triangle ABC is given with θ = 1.1 and |AC| = b = 4. CD is drawn perpendicular to AB, DE is
drawn perpendicular to BC, EF
AB and this process is continued indefinitely as shown in the
figure. Find the total length of all the perpendiculars |CD| + |DE| + |EF| + |FG| + ... Stewart - Calculus 6e Chapter 12 Form F ∞ 7. Let A = ∑ n =10 3
n 4.6 ∞ and B = ∫
9 3 dx . Compare A and B. x 4.6 Select the correct answer.
a.
8. A ≥ B b. A ≤ B c. A < B d. A > B e. A = B Given the two series
A = 1+ 1
1
1
1
+ +
+
+ ... and B =
16 81 256 625 ∞ ∑ n8e − n 9 determine whether each series is convergent n =1 or divergent and choose the correct statement.
Select the correct answer.
a.
b.
c.
d.
9. Series A is divergent, series B is convergent.
Both series are convergent.
Series A is convergent, series B is divergent.
Both series are divergent. Find the values of s for which the series is convergent.
∞ ∑
k =5 1
m ln m[ln(ln m)] s ∞ 10. Find the partial sum s10 of the series ∑
n =1 7
.
n3 Please approximate the answer to the nearest thousandth.
∞ 11. Find the partial sum s7 of the series ∑ m =1 8
.
6 + 10 m Give your answer to five decimal places.
12. Use the sum of the first 9 terms to approximate the sum of the following series:
∞ ∑
n =1 4
n7 + n5 Give your answer to six decimal places.
13. Which of the given series is (are) convergent? Select the correct answer.
a.
b. 7 7 7 7 7
− + − + −. . .
6 7 8 9 10
1 2 3 4 5
− + − + − +. . .
6 7 8 9 10 Stewart - Calculus 6e Chapter 12 Form F 14. Test the series for convergence or divergence.
6
6
6
6
6
−
+
−
+
−. . .
ln 2 ln 3 ln 4 ln 5 ln 6 Select the correct answer.
a. the series is divergent
b. the series is convergent
15. Approximate the sum to the indicated accuracy.
∞ ∑
n=0 (−1) n
(four decimal places)
3n n ! 16. The terms of a series are defined recursively by the equations a1 = 5, an +1 = Determine whether ∑a n 7n + 1
an
6n + 3 converges or diverges. 17. Estimate sin(33D ) correct to five decimal places. Select the correct answer.
a. 1.54464 b. 1.04464 c. 0.54464 d. 2.04464 18. Test the series for convergence or divergence. ∑(
∞ k ) 3 −1 k k =1 ∞ 19. For which positive integers k is the series ∑
n =1 (n!) 6
convergent?
(kn)! Select the correct answer.
a.
b.
c.
d.
e. k ≥1
k≤0
k≥0
k≥6
k ≤ −6 20. Find the Taylor polynomial T3 for the function f (x) at the number a = 1.
f ( x) = sin x e. 3.54464 ANSWER KEY
Stewart - Calculus 6e Chapter 12 Form F 1. an = 6n − 5 2. 0 3. a 4. d 5. 0.0428 6. 32.77 7. c 8. b 9. (1, ∞) 10. 8.383
11. 0.58431
12. 2.026946
13. a
14. b
15. 0.7165
16. diverges
17. c
18. convergent
19. d
20. T3 = sin(1) + cos(1) ⋅ ( x − 1) − cos(1)
sin(1)
⋅ ( x − 1) 2 −
⋅ ( x − 1) 3
6
2 Stewart - Calculus 6e Chapter 12 Form G 1. Find the value of the limit for the sequence.
⎧⎪ n8 ⎫⎪
⎨ ⎬
⎪⎩ n! ⎪⎭ 2. Find the exact value of the limit of the sequence defined by a1 = 3 , an +1 = 3 + an . 3. Find the exact value of the limit of the sequence defined by a1 = 1, an +1 = 6 − 4. A right triangle ABC is given with θ = 1.4 and |AC| = b = 10. CD is drawn perpendicular to AB, DE
is drawn perpendicular to BC, EF
AB and this process is continued indefinitely as shown in the 1
an figure. Find the total length of all the perpendiculars |CD| + |DE| + |EF| + |FG| + ... 5. A sequence {an } is defined recursively by the equation an = 0.5(an −1 + an − 2 ) for n ≥ 3 where
a1 = 21, a2 = 21 . Use your calculator to guess the limit of the sequence. 6. Use the Integral Test to determine whether the series is convergent or divergent.
∞ ∑
n =1 1
9n + 1 Select the correct answer.
a.
b.
7. the series is divergent
the series is convergent Which of the following series is convergent?
Select the correct answer.
∞ a. ∑ m =1 10
m8 + 3 ∞ b. ∑ m =1 10 ln 6m
m2 ∞ c. ∑ m =1 5
m ln 3m Stewart - Calculus 6e Chapter 12 Form G ∞ 8. Find the partial sum s10 of the series ∑ m =1 5
.
m3 Please approximate the answer to the nearest thousandth.
∞ 9. Find all positive values of u for which the series ∑ 8u ln 5n converges. n =1 10. A car is moving with speed 16 m/s and acceleration 6m/s2 at a given instant. Using a second-degree
Taylor polynomial, estimate how far the car moves in the next second. Select the correct answer.
a. 19.5 m b. 19 m c. 20 m d. 25.5 m e. 30 m 11. Find the Taylor polynomial T3 for the function f at the number a = 1.
f ( x) = ln x Select the correct answer.
a.
b.
c.
d.
e. 1
1
( x + 1) − ( x + 1) 2 + ( x + 1) 3
2
3
1
1
2
( x − 1) − ( x − 1) + ( x − 1) 3
3
2
1
1
( x − 1) + ( x − 1) 2 + ( x − 1) 3
2
4
1
1
2
( x − 1) + ( x − 1) + ( x − 1)3
2
4
1
1
( x − 1) + ( x − 1) 2 + ( x − 1) 3
4
3 12. Use Taylor's Inequality to estimate the accuracy |R4| of the approximation f (x) at the number a = 1.5,
when 0 ≤ x ≤ 3 .
f ( x) = cos x 13. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?
∞ ∑
n =1 ( −1) n n
, ( | error | < 0.1562 )
2n Stewart - Calculus 6e Chapter 12 Form G 14. Approximate the sum to the indicated accuracy.
∞ ∑
n =1 (−1) n −1
n7 (five decimal places) Select the correct answer.
a. 0.99219 b. 0.99249 c. 0.97269 d. 0.99259
∞ 15. Which of the partial sums of the alternating series ∑
n =1 e. 0.98259 (−1) n −1
are overestimates of the total sum.
n Select the correct answer.
a. s100 b. s67 c. s82 d. s91 e. s55 16. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞ ∑
n =1 (−1) n −1
n4 n Select the correct answer.
a. divergent b. absolutely convergent c. conditionally convergent 17. Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
∞ ∑
n =1 (−1) n arctan n
n2 Select the correct answer.
a. absolutely convergent b. divergent c. conditionally convergent 18. Find the Maclaurin series for the function f (x).
f ( x)...

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