Question 1

Find the limit of the sequence {   }, where   =   -   .

Accepted Answer

A)  -54 
B)  54 
C)  -$\infty$ 
D)  0 
E) $\infty$ 
A)  -54 
B)  54 
C)  -$\infty$ 
D)  0 
E) $\infty$

Question 2

By possibly rearranging its terms, the series   can be forced to have the sum 17.

Accepted Answer

A) True 
 B)False

Question 3

How many terms of the Maclaurin series for   are needed to approximate   with error less than 0.00005 in size?

Accepted Answer

A)  4 
B)  5 
C)  6 
D)  7 
E)  3 
A)  4 
B)  5 
C)  6 
D)  7 
E)  3

Question 4

Give an upper bound for the error when   is approximated by the partial sum   .

Accepted Answer

A)  0.0004 
B)  0.0001 
C)  0.00004 
D)  0.00001 
E)  0.004 
A)  0.0004 
B)  0.0001 
C)  0.00004 
D)  0.00001 
E)  0.004

Question 5

Find all values of the nonzero constant real number p such that the series   is convergent.

Accepted Answer

A)  p > 1 
B)  0  0 
A)  p > 1 
B)  0  0

Question 6

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

Accepted Answer

A) True 
 B)False

Question 7

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

Accepted Answer

A)    
B)    
C)  1 
D)  $\infty$ (series diverges) 
E)  none of the above 
A)    
B)    
C)  1 
D)  $\infty$ (series diverges) 
E)  none of the above

Question 8

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

Question 9

Find the Fourier series for f(x) = x2 defined on [- $\pi$, $\pi$].

Accepted Answer

A)  f(x) =   -     cos(nx) 
B)  f(x) =   +     cos(nx) 
C)  f(x) =   +     cos(nx) 
D)  f(x) =   -     cos(nx) 
E)  f(x) =   +     sin(nx) 
A)  f(x) =   -     cos(nx) 
B)  f(x) =   +     cos(nx) 
C)  f(x) =   +     cos(nx) 
D)  f(x) =   -     cos(nx) 
E)  f(x) =   +     sin(nx)

Question 10

Find the sum   .

Accepted Answer

A)    
B)    
C)  1 
D)  $\infty$ (series diverges) 
E)  none of the above 
A)    
B)    
C)  1 
D)  $\infty$ (series diverges) 
E)  none of the above

Question 11

The geometric series

Accepted Answer

A)  converges and its sum is .   
B)  converges and its sum is .   
C)  diverges and hence it has no sum. 
D)  converges and its sum is 4. 
E)  converges and its sum is .   
A)  converges and its sum is .   
B)  converges and its sum is .   
C)  diverges and hence it has no sum. 
D)  converges and its sum is 4. 
E)  converges and its sum is .

Question 12

Find, if it exists, the limit of the sequence {   }, where an =   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
E)  The limit does not exist. 
A)  0 
B)  1 
C)  2 
D)  3 
E)  The limit does not exist.

Question 13

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

Accepted Answer

A)    
B)  cos(2) 
C)    
D)  cosh(2) + sinh(2) 
E)  -   
A)    
B)  cos(2) 
C)    
D)  cosh(2) + sinh(2) 
E)  -

Question 14

If {   } converges and {   } converges, then   must converge.

Accepted Answer

A) True 
 B)False

Question 15

Find the sum of the series

Accepted Answer

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

Question 16

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

Accepted Answer

A)  - cos(x), x =   
B)  cos(x), x =   
C)  sin(x), x =   
D)  sin(x), x =   
E)  none of the above 
A)  - cos(x), x =   
B)  cos(x), x =   
C)  sin(x), x =   
D)  sin(x), x =   
E)  none of the above

Question 17

If   is divergent, then   must be conditionally convergent.

Accepted Answer

A) True 
 B)False

Question 18

Accepted Answer

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

Question 19

The series   converges.

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

A)  centre 0, radius 1, interval [-1, 1] 
B)  centre 0, radius 1, interval (-1, 1] 
C)  centre 0, radius 1, interval (-1, 1) 
D)  centre 0, radius 2, interval (-2, 2) 
E)  centre 0, radius 1, interval [-1, 1) 
A)  centre 0, radius 1, interval [-1, 1] 
B)  centre 0, radius 1, interval (-1, 1] 
C)  centre 0, radius 1, interval (-1, 1) 
D)  centre 0, radius 2, interval (-2, 2) 
E)  centre 0, radius 1, interval [-1, 1)

Find the limit of the sequence { }, where = - .

By possibly rearranging its terms, the series can be forced to have the sum 17.

How many terms of the Maclaurin series for are needed to approximate with error less than 0.00005 in size?

Give an upper bound for the error when is approximated by the partial sum .

Find all values of the nonzero constant real number p such that the series is convergent.

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

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

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

Find the Fourier series for f(x) = x² defined on [- $\pi$ , $\pi$ ].

Find the sum .

The geometric series

Find, if it exists, the limit of the sequence { }, where a_n = .

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

If { } converges and { } converges, then must converge.

Find the sum of the series

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

If is divergent, then must be conditionally convergent.

The series converges.

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

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 limit of the sequence { }, where = - .

By possibly rearranging its terms, the series can be forced to have the sum 17.

How many terms of the Maclaurin series for are needed to approximate with error less than 0.00005 in size?

Give an upper bound for the error when is approximated by the partial sum .

Find all values of the nonzero constant real number p such that the series is convergent.

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

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

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

Find the Fourier series for f(x) = x2 defined on [- π\piπ , π\piπ ].

Find the sum .

The geometric series

Find, if it exists, the limit of the sequence { }, where an = .

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

If { } converges and { } converges, then must converge.

Find the sum of the series

Assuming that no calculator is handy, what known Maclaurin series would you use to approximate sin(95°) using as few terms as possible, and at what point would you evaluate that series?

If is divergent, then must be conditionally convergent.

The series converges.

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

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

Find the Fourier series for f(x) = x² defined on [- $\pi$ , $\pi$ ].

Find, if it exists, the limit of the sequence { }, where a_n = .