Question 1

Find the limit of the sequence   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
E)  This sequence is divergent. 
A)  0 
B)  1 
C)  2 
D)  3 
E)  This sequence is divergent.

Question 2

Find the sum of the series   .

Accepted Answer

A)  2 
B)  3 
C)  1 
D)  4 
E)  $\infty$ (series diverges) 
A)  2 
B)  3 
C)  1 
D)  4 
E)  $\infty$ (series diverges)

Question 3

How many nonzero terms of the Maclaurin series for   (x) are needed to approximate   (0.2) correct to 5 decimal places?

Accepted Answer

A)  2 
B)  3 
C)  4 
D)  5 
E)  1 
A)  2 
B)  3 
C)  4 
D)  5 
E)  1

Question 4

Find the radius, centre, 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 $\infty$, interval (-$\infty$, $\infty$) 
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 $\infty$, interval (-$\infty$, $\infty$) 
E)  centre 0, radius 1, interval (-1, 1]

Question 5

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

Accepted Answer

A)  p $\ge$1 
B)  p > 1 
C)  p $\ge$ 2 
D)  p > 2 
E)  p > 0 
A)  p $\ge$1 
B)  p > 1 
C)  p $\ge$ 2 
D)  p > 2 
E)  p > 0

Question 6

Evaluate   .

Accepted Answer

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

Question 7

The series   converges.

Accepted Answer

A) True 
 B)False

Question 8

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

Accepted Answer

A)  centre 1, radius   , interval     
B)  centre 1, radius   , interval     
C)  centre 1, radius   , interval     
D)  centre 1, radius   , interval     
E)  centre 1, radius   , interval   ,   
A)  centre 1, radius   , interval     
B)  centre 1, radius   , interval     
C)  centre 1, radius   , interval     
D)  centre 1, radius   , interval     
E)  centre 1, radius   , interval   ,

Question 9

Find the limit of the sequence   .

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  3 
E)  6 
A)  0 
B)  1 
C)  2 
D)  3 
E)  6

Question 10

If   and   both diverge, then   also diverges.

Accepted Answer

A) True 
 B)False

Question 11

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

Question 12

On what interval of values of x does the series   converge? What is its sum for x in that interval?

Accepted Answer

A)  (-1, 1), sum =   
B)  (-1, 1), sum =   
C)  (-1, 1], sum =   
D)  [-1, 1), sum =   
E)  [-1, 1], sum =   
A)  (-1, 1), sum =   
B)  (-1, 1), sum =   
C)  (-1, 1], sum =   
D)  [-1, 1), sum =   
E)  [-1, 1], sum =

Question 13

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

Accepted Answer

A)  2 
B)  3 
C)  6 
D)  9 
E)  4 
A)  2 
B)  3 
C)  6 
D)  9 
E)  4

Question 14

Find the Taylor expansion of 4   + 3   + 2   + y + 2 in powers of y + 2.

Accepted Answer

A)  48 + 99(y + 2) + 80   - 29   +4   
B)  48 - 99(y + 2) - 80   - 29   + 4   
C)  48 + 99(y + 2) + 80   + 29   + 4   
D)  48 - 99(y + 2) + 80   - 29   + 4   
E)  -48 + 99(y + 2) - 80   + 29   - 4   
A)  48 + 99(y + 2) + 80   - 29   +4   
B)  48 - 99(y + 2) - 80   - 29   + 4   
C)  48 + 99(y + 2) + 80   + 29   + 4   
D)  48 - 99(y + 2) + 80   - 29   + 4   
E)  -48 + 99(y + 2) - 80   + 29   - 4

Question 15

Using the known geometric series representation   =     , valid for -1 < x < 1, find a power series representation for f(x) = ln(2 + x) in powers of x - 1. On what interval does the series converge to f(x)?

Accepted Answer

A)  f(x) = ln 3 +       , for -0 < x $\le$ 2 
B)  f(x) = ln 3 +       , for -2 < x $\le$ 4 
C)  f(x) = ln 2 +       , for -1 < x $\le$ 3 
D)  f(x) = ln 2 +       , for -1 < x < 3 
E)  none of the above 
A)  f(x) = ln 3 +       , for -0 < x $\le$ 2 
B)  f(x) = ln 3 +       , for -2 < x $\le$ 4 
C)  f(x) = ln 2 +       , for -1 < x $\le$ 3 
D)  f(x) = ln 2 +       , for -1 < x < 3 
E)  none of the above

Question 16

Find the first three nonzero terms in the Maclaurin series for tan(2x).

Accepted Answer

A)  2x +     +     
B)  2x -     +     
C)  2x +     +     
D)  2x -     +     
E)  2x +     -     
A)  2x +     +     
B)  2x -     +     
C)  2x +     +     
D)  2x -     +     
E)  2x +     -

Question 17

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

Accepted Answer

A)  centre -3, radius 1, interval [-4, -2] 
B)  centre -3, radius 1, interval (-4, -2) 
C)  centre 0, radius $\infty$, interval (-$\infty$, $\infty$) 
D)  centre -3, radius $\infty$, interval (-$\infty$, $\infty$) 
E)  centre -3, radius    , interval   
A)  centre -3, radius 1, interval [-4, -2] 
B)  centre -3, radius 1, interval (-4, -2) 
C)  centre 0, radius $\infty$, interval (-$\infty$, $\infty$) 
D)  centre -3, radius $\infty$, interval (-$\infty$, $\infty$) 
E)  centre -3, radius    , interval

Question 18

What is the largest positive constant K such that if 0 < p < K, then   must converge?

Accepted Answer

A)  1 
B)  2 
C)  4 
D)  8 
E)  9 
A)  1 
B)  2 
C)  4 
D)  8 
E)  9

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

For what values of x does the series     
(a) converge absolutely? 
(b) converge conditionally?

Accepted Answer

A)  (a) on the interval (1, 5), $~~~~~~~~$(b) at x = 1 
B)  (a) on the interval (1, 5), $~~~~~~~~$(b) at x = -1 and x = 5 
C)  (a) on the interval [1, 5], $~~~~~~~~$(b) nowhere 
D)  (a) on the interval [1, 5), $~~~~~~~~$(b) at x = 5 
E)  (a) all real x, $~~~~~~~~$(b) nowhere 
A)  (a) on the interval (1, 5), $~~~~~~~~$(b) at x = 1 
B)  (a) on the interval (1, 5), $~~~~~~~~$(b) at x = -1 and x = 5 
C)  (a) on the interval [1, 5], $~~~~~~~~$(b) nowhere 
D)  (a) on the interval [1, 5), $~~~~~~~~$(b) at x = 5 
E)  (a) all real x, $~~~~~~~~$(b) nowhere

Find the limit of the sequence .

Find the sum of the series .

How many nonzero terms of the Maclaurin series for (x) are needed to approximate (0.2) correct to 5 decimal places?

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

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

Evaluate .

The series converges.

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

Find the limit of the sequence .

If and both diverge, then also diverges.

On what interval of values of x does the series converge? What is its sum for x in that interval?

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

Find the Taylor expansion of 4 + 3 + 2 + y + 2 in powers of y + 2.

Using the known geometric series representation = , valid for -1 < x < 1, find a power series representation for f(x) = ln(2 + x) in powers of x - 1. On what interval does the series converge to f(x)?

Find the first three nonzero terms in the Maclaurin series for tan(2x).

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

What is the largest positive constant K such that if 0 < p < K, then must converge?

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

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

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 .

Find the sum of the series .

How many nonzero terms of the Maclaurin series for (x) are needed to approximate (0.2) correct to 5 decimal places?

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

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

Evaluate .

The series converges.

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

Find the limit of the sequence .

If and both diverge, then also diverges.

On what interval of values of x does the series converge? What is its sum for x in that interval?

How many nonzero terms of the known Maclaurin series for ln(1 + x) are needed to approximate ln(1.1) correct to 4 decimal places?

Find the Taylor expansion of 4 + 3 + 2 + y + 2 in powers of y + 2.

Using the known geometric series representation = , valid for -1 < x < 1, find a power series representation for f(x) = ln(2 + x) in powers of x - 1. On what interval does the series converge to f(x)?

Find the first three nonzero terms in the Maclaurin series for tan(2x).

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

What is the largest positive constant K such that if 0 < p < K, then must converge?

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

For what values of x does the series (a) converge absolutely? (b) converge conditionally?

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