Question 1

Find   (0) if f(x) =   .

Accepted Answer

A)    
B)    
C)    
D)    
E)  1 
A)    
B)    
C)    
D)    
E)  1 C

Question 2

Find the Maclaurin series (binomial series) for   (x).

Accepted Answer

A)  x -   for -1 < x < 1 
B)  1 -   for -1 < x < 1 
C)  x +   for -1 < x < 1 
D)  x +   for -1 < x < 1 
E)  1 +   for -1 < x < 1 
A)  x -   for -1 < x < 1 
B)  1 -   for -1 < x < 1 
C)  x +   for -1 < x < 1 
D)  x +   for -1 < x < 1 
E)  1 +   for -1 < x < 1 C

Question 3

The radius of convergence of the power series    
Which of the following statements is true?

Accepted Answer

A)  The power series converges for all x   (- $\infty$ , $\infty$). 
B)  The power series converges only at x = - 4. 
C)  The power series converges for all x   (- $\infty$ , 4)?(4 , $\infty$). 
D)  The power series diverges for all x    (- $\infty$ , $\infty$). 
E)  The power series converges only at x = 4. 
A)  The power series converges for all x   (- $\infty$ , $\infty$). 
B)  The power series converges only at x = - 4. 
C)  The power series converges for all x   (- $\infty$ , 4)?(4 , $\infty$). 
D)  The power series diverges for all x    (- $\infty$ , $\infty$). 
E)  The power series converges only at x = 4. E

Question 4

Given f(x) = ln(3x + 1)
(a) Find Taylor's polynomial of degree n = 2 for f about c = 0
(b) Use part (a) to estimate the value of ln(1.3).
(c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

Accepted Answer

(a) @#IMG-DLM& (x) = 3x - @#IMG-DLM& @#IMG-DLM&

Question 5

Find the limit of the sequence   .

Accepted Answer

A)  9 
B)  3 
C)  -3 
D)  0 
E)  -9 
A)  9 
B)  3 
C)  -3 
D)  0 
E)  -9

Question 6

Accepted Answer

A)  f(x) =   , for -1 < x $\le$ 1 
B)  f(x) =   , for -1 < x < 1 
C)  f(x) =   , for -1 $\le$ x $\le$ 1 
D)  f(x) =   , for -1 $\le$ x < 1 
E)  f(x) =   , for -1 < x $\le$ 1 
A)  f(x) =   , for -1 < x $\le$ 1 
B)  f(x) =   , for -1 < x < 1 
C)  f(x) =   , for -1 $\le$ x $\le$ 1 
D)  f(x) =   , for -1 $\le$ x < 1 
E)  f(x) =   , for -1 < x $\le$ 1

Question 7

Which of the three series (i)   (ii)   (iii)   is absolutely convergent?

Accepted Answer

A)  series (i) and (ii) 
B)  series (i) 
C)  series (ii) and (iii) 
D)  series (i) and (iii) 
E)  none of the series 
A)  series (i) and (ii) 
B)  series (i) 
C)  series (ii) and (iii) 
D)  series (i) and (iii) 
E)  none of the series

Question 8

Accepted Answer

A) True 
 B)False

Question 9

Which of the following descriptors apply to the sequence   ?
(a) increasing (or ultimately increasing)
(b) decreasing (or ultimately decreasing)
(c) positive (or ultimately positive)
(d) negative (or ultimately negative)
(e) bounded below only
(f) bounded above only
(g) bounded
(h) unbounded above and below
(i) alternating
(j) divergent (but not to $\infty$or -$\infty$)
(k) divergent to $\infty$
(l) divergent to -$\infty$
(m) convergent

Accepted Answer

A)  (g), (i), (m) 
B)  (a), (f), (m) 
C)  (g), (i), (j) 
D)  (b), (c), (i), (l) 
E)  (g), (i), (k) 
A)  (g), (i), (m) 
B)  (a), (f), (m) 
C)  (g), (i), (j) 
D)  (b), (c), (i), (l) 
E)  (g), (i), (k)

Question 10

Find the Taylor expansion of   about x = b.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

The sequence

Accepted Answer

A)  diverges to - $\infty$. 
B)  converges to 0. 
C)  converges to 3. 
D)  diverges to + $\infty$. 
E)  converges to - 4. 
A)  diverges to - $\infty$. 
B)  converges to 0. 
C)  converges to 3. 
D)  diverges to + $\infty$. 
E)  converges to - 4.

Question 12

(x) = x +   for -1 < x < 1

Accepted Answer

A) True 
 B)False

Question 13

The series   converges.

Accepted Answer

A) True 
 B)False

Question 14

For what values of x does the series   converge?

Accepted Answer

A)  on the intervals (-$\infty$, -1] and [1, $\infty$) 
B)  on the intervals (-$\infty$, -1) and (1, $\infty$) 
C)  on the interval [-1, 1] 
D)  on the interval (-1, 1) 
E)  on the interval [1, $\infty$) 
A)  on the intervals (-$\infty$, -1] and [1, $\infty$) 
B)  on the intervals (-$\infty$, -1) and (1, $\infty$) 
C)  on the interval [-1, 1] 
D)  on the interval (-1, 1) 
E)  on the interval [1, $\infty$)

Question 15

For what values of x does the series ln x +   +   +   +... converge?

Accepted Answer

A)    < x < e 
B)  0 < x < e 
C)  1 < x < e 
D)  0 < x < $\infty$ 
E)  0 < x < 1 
A)    < x < e 
B)  0 < x < e 
C)  1 < x < e 
D)  0 < x < $\infty$ 
E)  0 < x < 1

Question 16

Use the integral test bounds to estimate the sum of the series   using the first three terms.

Accepted Answer

A)  104 
B)  140 
C)  119 
D)  101 
E)  98 
A)  104 
B)  140 
C)  119 
D)  101 
E)  98

Question 17

The series   converges.

Accepted Answer

A) True 
 B)False

Question 18

Find the centre, radius, and interval of convergence of the series

Accepted Answer

A)  centre 4, radius 3, interval (1, 7] 
B)  centre 4, radius 3, interval [1, 7) 
C)  centre 4, radius 1, interval [3, 5] 
D)  centre 4, radius 1, interval (3, 5) 
E)  centre 4, radius 1, interval (3, 5] 
A)  centre 4, radius 3, interval (1, 7] 
B)  centre 4, radius 3, interval [1, 7) 
C)  centre 4, radius 1, interval [3, 5] 
D)  centre 4, radius 1, interval (3, 5) 
E)  centre 4, radius 1, interval (3, 5]

Question 19

Show that the sequence   converges, and find its limit.

Accepted Answer

The answer of Show that the sequence   converges,...

Question 20

The series   converges.

Accepted Answer

A) True 
 B)False

Find (0) if f(x) = .

Find the Maclaurin series (binomial series) for (x).

The radius of convergence of the power series Which of the following statements is true?

Given f(x) = ln(3x + 1) (a) Find Taylor's polynomial of degree n = 2 for f about c = 0 (b) Use part (a) to estimate the value of ln(1.3). (c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

Find the limit of the sequence .

Which of the three series (i) (ii) (iii) is absolutely convergent?

Find the Taylor expansion of about x = b.

The sequence

(x) = x + for -1 < x < 1

The series converges.

For what values of x does the series converge?

For what values of x does the series ln x + + + +... converge?

Use the integral test bounds to estimate the sum of the series using the first three terms.

The series converges.

Find the centre, radius, and interval of convergence of the series

Show that the sequence converges, and find its limit.

The series converges.

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Exam 10: Sequences, Series, and Power Series

Find (0) if f(x) = .

Find the Maclaurin series (binomial series) for (x).

The radius of convergence of the power series Which of the following statements is true?

Given f(x) = ln(3x + 1) (a) Find Taylor's polynomial of degree n = 2 for f about c = 0 (b) Use part (a) to estimate the value of ln(1.3). (c) Estimate the maximum absolute error involved in the approximation obtained in part (b) above.

Find the limit of the sequence .

Which of the three series (i) (ii) (iii) is absolutely convergent?

Find the Taylor expansion of about x = b.

The sequence

(x) = x + for -1 < x < 1

The series converges.

For what values of x does the series converge?

For what values of x does the series ln x + + + +... converge?

Use the integral test bounds to estimate the sum of the series using the first three terms.

The series converges.

Find the centre, radius, and interval of convergence of the series

Show that the sequence converges, and find its limit.

The series converges.

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters