Question 1

The interval of convergence of the power series   is given by [-7, -3).Find the centre c and the radius of convergence R of the power series.

Accepted Answer

A)  c = -10, R = 4 
B)  c = -5, R = 2 
C)  c = -4, R = 10 
D)  c = -2, R = 5 
E)  c = 2, R = -5 
A)  c = -10, R = 4 
B)  c = -5, R = 2 
C)  c = -4, R = 10 
D)  c = -2, R = 5 
E)  c = 2, R = -5

Question 2

If {|   |} converges, then {   } converges.

Accepted Answer

A) True 
 B)False

Question 3

Find the Maclaurin series for f(x) =   .

Accepted Answer

A)  f(x) =   
B)  f(x) =   
C)  f(x) =   
D)  f(x) =   
E)  f(x) =   
A)  f(x) =   
B)  f(x) =   
C)  f(x) =   
D)  f(x) =   
E)  f(x) =

Question 4

Use known Maclaurin series to evaluate .

Accepted Answer

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

Question 5

Find the Taylor series for f(x) = ln   about c = - 3. Where does the series converge to f(x)?

Accepted Answer

A)  - 30-   , -   < x $\le$   
B)  ln(6) +   ,   < x $\le$   
C)  ln(6) -   , -   < x $\le$ -   
D)  ln(6) -   , -   $\le$ x < -   
E)  ln(6) +   ,   $\le$ x <   
A)  - 30-   , -   < x $\le$   
B)  ln(6) +   ,   < x $\le$   
C)  ln(6) -   , -   < x $\le$ -   
D)  ln(6) -   , -   $\le$ x < -   
E)  ln(6) +   ,   $\le$ x <

Question 6

For exactly what values of the constant p does the series   converge?

Accepted Answer

A)  -3 < p < 3 
B)  -3 $\le$ p $\le$ 3 
C)  0 $\le$ p $\le$ 3 
D)  0 < p < 3 
E)  -3 < p < 0 
A)  -3 < p < 3 
B)  -3 $\le$ p $\le$ 3 
C)  0 $\le$ p $\le$ 3 
D)  0 < p < 3 
E)  -3 < p < 0

Question 7

Find the sum of the series 4 - 1 +   -   +...

Accepted Answer

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

Question 8

The series   converges.

Accepted Answer

A) True 
 B)False

Question 9

Find the Maclaurin series for ln   . For what values of x does the series converge to the function?

Accepted Answer

A)    , for -1 < x < 1 
B)    , for -1 < x < 1 
C)    , for -1 < x < 1 
D)    , for -1 < x < 1 
E)    , for all x 
A)    , for -1 < x < 1 
B)    , for -1 < x < 1 
C)    , for -1 < x < 1 
D)    , for -1 < x < 1 
E)    , for all x

Question 10

Use known Maclaurin series to evaluate .

Accepted Answer

A)  -20 
B)  20 
C)  -30 
D)  30 
E)  10 
A)  -20 
B)  20 
C)  -30 
D)  30 
E)  10

Question 11

The series   converges.

Accepted Answer

A) True 
 B)False

Question 12

Let f(x) denote the sum of the series   wherever the series converges. Where does the series converge? Calculate   (x) and f(0). What do these results imply that f(x) actually is?

Accepted Answer

Series converges to

Question 13

Which of the following descriptors apply to the sequence   ?

Accepted Answer

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

Question 14

Find the Maclaurin series (binomial series) for   .

Accepted Answer

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

Question 15

If {   } converges, then   must diverge.

Accepted Answer

A) True 
 B)False

Question 16

Find the limit of the sequence   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
E)    
A)  0 
B)  1 
C)  2 
D)  3 
E)

Question 17

Determine all values of the constant real number k so that the Fourier series of the periodic function f(t) =   , f(t + 4) = f(t)
Converges to f(t) for all t    (- $\infty$, $\infty$).

Accepted Answer

A)  only -2 
B)  6 and -5 
C)  - 2 and 6 
D)  only 6 
E)  -2 , -5 and 6 
A)  only -2 
B)  6 and -5 
C)  - 2 and 6 
D)  only 6 
E)  -2 , -5 and 6

Question 18

Find the sum of the series   .

Accepted Answer

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

Question 19

If   converges, then     = 0.

Accepted Answer

A) True 
 B)False

Question 20

The sequence   is defined recursively by   =   ,   =   , n = 1, 2, 3,...Assuming the sequence converges to the real number L, find L.

Accepted Answer

A)  1 
B)  -4 or 1 
C)  -1 or 4 
D)  -1 
E)  4 
A)  1 
B)  -4 or 1 
C)  -1 or 4 
D)  -1 
E)  4

The interval of convergence of the power series is given by [-7, -3).Find the centre c and the radius of convergence R of the power series.

If {| |} converges, then { } converges.

Find the Maclaurin series for f(x) = .

Use known Maclaurin series to evaluate .

Find the Taylor series for f(x) = ln about c = - 3. Where does the series converge to f(x)?

For exactly what values of the constant p does the series converge?

Find the sum of the series 4 - 1 + - +...

The series converges.

Find the Maclaurin series for ln . For what values of x does the series converge to the function?

Use known Maclaurin series to evaluate .

The series converges.

Let f(x) denote the sum of the series wherever the series converges. Where does the series converge? Calculate (x) and f(0). What do these results imply that f(x) actually is?

Which of the following descriptors apply to the sequence ?

Find the Maclaurin series (binomial series) for .

If { } converges, then must diverge.

Find the limit of the sequence .

Find the sum of the series .

If converges, then = 0.

The sequence is defined recursively by = , = , n = 1, 2, 3,...Assuming the sequence converges to the real number L, find L.

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

Exam 10: Sequences, Series, and Power Series

The interval of convergence of the power series is given by [-7, -3).Find the centre c and the radius of convergence R of the power series.

If {| |} converges, then { } converges.

Find the Maclaurin series for f(x) = .

Use known Maclaurin series to evaluate .

Find the Taylor series for f(x) = ln about c = - 3. Where does the series converge to f(x)?

For exactly what values of the constant p does the series converge?

Find the sum of the series 4 - 1 + - +...

The series converges.

Find the Maclaurin series for ln . For what values of x does the series converge to the function?

Use known Maclaurin series to evaluate .

The series converges.

Let f(x) denote the sum of the series wherever the series converges. Where does the series converge? Calculate (x) and f(0). What do these results imply that f(x) actually is?

Which of the following descriptors apply to the sequence ?

Find the Maclaurin series (binomial series) for .

If { } converges, then must diverge.

Find the limit of the sequence .

Determine all values of the constant real number k so that the Fourier series of the periodic function f(t) = , f(t + 4) = f(t) Converges to f(t) for all t (- ∞\infty∞ , ∞\infty∞ ).

Find the sum of the series .

If converges, then = 0.

The sequence is defined recursively by = , = , n = 1, 2, 3,...Assuming the sequence converges to the real number L, find L.

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