Question 1

Find the Fourier series of the function f(x) = x defined on the interval [-1, 1].

Accepted Answer

A)    
B)    
C)    
D)    
E)  none of the above 
A)    
B)    
C)    
D)    
E)  none of the above

Question 2

=   is the   partial sum of an infinite series    . Find   . Determine if the series is convergent or divergent, and if convergent find its sum.

Accepted Answer

A)    =   , series diverges to $\infty$ 
B)    =   , series diverges to $\infty$ 
C)    =   , series converges to 1 
D)    =   , series converges to 2 
E)    =   , series converges to 1 
A)    =   , series diverges to $\infty$ 
B)    =   , series diverges to $\infty$ 
C)    =   , series converges to 1 
D)    =   , series converges to 2 
E)    =   , series converges to 1

Question 3

The series   converges.

Accepted Answer

A) True 
 B)False

Question 4

Find the Maclaurin series (binomial series) for   .

Accepted Answer

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

Question 5

=   is the   partial sum of an infinite series   . Find   . Determine if the series is convergent or divergent, and if convergent find its sum.

Accepted Answer

A)    =   ,   = -   for n $\ge$ 2; series converges, sum is 0 
B)    =   for n $\ge$ 1; series converges, sum is 1 
C)    =   ,   = -   for n $\ge$ 2; series converges, sum is   
D)    =   for n $\ge$1; series converges, sum is   
E)    =   for n $\ge$ 1; series converges, sum is   
A)    =   ,   = -   for n $\ge$ 2; series converges, sum is 0 
B)    =   for n $\ge$ 1; series converges, sum is 1 
C)    =   ,   = -   for n $\ge$ 2; series converges, sum is   
D)    =   for n $\ge$1; series converges, sum is   
E)    =   for n $\ge$ 1; series converges, sum is

Question 6

The error E in the approximation   $\approx$  satisfies   <   .

Accepted Answer

A) True 
 B)False

Question 7

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

Accepted Answer

A)  series (iii) 
B)  series (i) 
C)  series (i) and (iii) 
D)  series (ii) 
E)  All three 
A)  series (iii) 
B)  series (i) 
C)  series (i) and (iii) 
D)  series (ii) 
E)  All three

Question 8

Find the Maclaurin series for the function F(x) =   dt.

Accepted Answer

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

Question 9

By suitably manipulating the geometric series   =   , -1 < x < 1, you can calculate the sum of the series   . Use this method to evaluate the sum   .

Accepted Answer

A)  4 
B)  6 
C)  8 
D)  10 
E)  none of the above 
A)  4 
B)  6 
C)  8 
D)  10 
E)  none of the above

Question 10

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

Accepted Answer

A)  p > 2 
B)  p $\ge$ 2 
C)  p  1 
A)  p > 2 
B)  p $\ge$ 2 
C)  p  1

Question 11

Find the sum of the series   +   +   +... +   +... =   .

Accepted Answer

A)    
B)  1 
C)  2 
D)    
E)  $\infty$ (series diverges) 
A)    
B)  1 
C)  2 
D)    
E)  $\infty$ (series diverges)

Question 12

Find the sum of the series   .

Accepted Answer

A)  sin(2) - 1 
B)  sin(2) -   
C)    
D)    
E)  cos(2) -   
A)  sin(2) - 1 
B)  sin(2) -   
C)    
D)    
E)  cos(2) -

Question 13

If f(x) =   x, find   (0).

Accepted Answer

A)  -98! 
B)  98! 
C)  -99! 
D)  99! 
E)  0 
A)  -98! 
B)  98! 
C)  -99! 
D)  99! 
E)  0

Question 14

The series   converges.

Accepted Answer

A) True 
 B)False

Question 15

The series   converges.

Accepted Answer

A) True 
 B)False

Question 16

Show that the infinite series    converges, and find its sum.Hint : Verify that = - .

Accepted Answer

The answer of Show that the infinite series  ...

Question 17

Calculate the Maclaurin series for the function G(x) =   and calculate G(1) correct to 5 decimal places.

Accepted Answer

A)  G(x) =   , G(1)$\approx$ 0.31025 
B)  G(x) =   , G(1) $\approx$ 0.31026 
C)  G(x) =   , G(1) $\approx$0.31029 
D)  G(x) =   , G(1) $\approx$ 0.31031 
E)  G(x) =   , G(1) $\approx$ 0.31024 
A)  G(x) =   , G(1)$\approx$ 0.31025 
B)  G(x) =   , G(1) $\approx$ 0.31026 
C)  G(x) =   , G(1) $\approx$0.31029 
D)  G(x) =   , G(1) $\approx$ 0.31031 
E)  G(x) =   , G(1) $\approx$ 0.31024

Question 18

Find the Maclaurin series for f(x) =   x. For what values of x does it converge to f(x)?

Accepted Answer

A)  f(x) =     ,for all x 
B)  f(x) =     ,for all x 
C)  f(x) =       , for all x 
D)  f(x) =       , for all x 
E)  f(x) =       , for all x 
A)  f(x) =     ,for all x 
B)  f(x) =     ,for all x 
C)  f(x) =       , for all x 
D)  f(x) =       , for all x 
E)  f(x) =       , for all x

Question 19

The series   converges.

Accepted Answer

A) True 
 B)False

Question 20

Find the Taylor series for f(x) =   about x - 3. On what interval does the series converge to f(x)?

Accepted Answer

A)  f(x) =   , for 0 < x < 6 
B)  f(x) =   , for 0 < x < 6 
C)  f(x) =   , for 0 < x $\le$ 6 
D)  f(x) =   , for 0 < x < 6 
E)  f(x) =   , for 0 < x $\le$ 6 
A)  f(x) =   , for 0 < x < 6 
B)  f(x) =   , for 0 < x < 6 
C)  f(x) =   , for 0 < x $\le$ 6 
D)  f(x) =   , for 0 < x < 6 
E)  f(x) =   , for 0 < x $\le$ 6

Find the Fourier series of the function f(x) = x defined on the interval [-1, 1].

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

The series converges.

Find the Maclaurin series (binomial series) for .

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

The error E in the approximation $The error E in the approximation \approx satisfies < .$ $\approx$ $The error E in the approximation \approx satisfies < .$ satisfies $The error E in the approximation \approx satisfies < .$ < $The error E in the approximation \approx satisfies < .$ .

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

Find the Maclaurin series for the function F(x) = dt.

By suitably manipulating the geometric series = , -1 < x < 1, you can calculate the sum of the series . Use this method to evaluate the sum .

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

Find the sum of the series + + +... + +... = .

Find the sum of the series .

If f(x) = x, find (0).

The series converges.

The series converges.

Show that the infinite series converges, and find its sum.Hint : Verify that = - .

Calculate the Maclaurin series for the function G(x) = and calculate G(1) correct to 5 decimal places.

Find the Maclaurin series for f(x) = x. For what values of x does it converge to f(x)?

The series converges.

Find the Taylor series for f(x) = about x - 3. On what interval does the series converge to f(x)?

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

Find the Fourier series of the function f(x) = x defined on the interval [-1, 1].

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

The series converges.

Find the Maclaurin series (binomial series) for .

= is the partial sum of an infinite series . Find . Determine if the series is convergent or divergent, and if convergent find its sum.

The error E in the approximation ≈\approx≈ satisfies < .

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

Find the Maclaurin series for the function F(x) = dt.

By suitably manipulating the geometric series = , -1 < x < 1, you can calculate the sum of the series . Use this method to evaluate the sum .

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

Find the sum of the series + + +... + +... = .

Find the sum of the series .

If f(x) = x, find (0).

The series converges.

The series converges.

Show that the infinite series converges, and find its sum.Hint : Verify that = - .

Calculate the Maclaurin series for the function G(x) = and calculate G(1) correct to 5 decimal places.

Find the Maclaurin series for f(x) = x. For what values of x does it converge to f(x)?

The series converges.

Find the Taylor series for f(x) = about x - 3. On what interval does the series converge to f(x)?

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

The error E in the approximation $The error E in the approximation \approx satisfies < .$ $\approx$ $The error E in the approximation \approx satisfies < .$ satisfies $The error E in the approximation \approx satisfies < .$ < $The error E in the approximation \approx satisfies < .$ .