AP CALCULUS BC Scoring Guide

9.1

1. A particle moves in the xy-plane so that its position for t ≥ 0 is given by the parametric equations x = ln (t + 1) and y
= kt2 , where k is a positive constant. The line tangent to the particle’s path at the point where t = 3 has slope 8.
What is the value of k ?
(A)

(B)

(C)
(D)

2. For an object travels along an elliptical path given by the parametric equations and
. At the point where the object leaves the path and travels along the line tangent to the path at
that point. What is the slope of the line on which the object travels?
(A)
(B)
(C)

(D)

(E)

3. For what values of t does the curve given by the parametric equations and
have a vertical tangent?
(A) 0 only
(B) 1 only
(C) 0 and only

(D) 0, , and 1
(E) No value

4. If x=t2-1 and y=2et, then

(A)

(B)

(C)

(D)
(E) et

5. A curve is described by the parametric equations x = t2 + 2t and y = t3 + t2. An equation of the line tangent to the
curve at the point determined by t = 1 is

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9.1

(A) 2x - 3y = 0
(B) 4x - 5y = 2
(C) 4x - y = 10
(D) 5x - 4y = 7
(E) 5x - y = 13

6. A curve in the plane is defined parametrically by the equations and An equation of the
line tangent to the curve at t=1 is
(A) y=2x
(B) y=8x
(C) y=2x-1
(D) y=4x-5
(E) y=8x+13

7. A curve C is defined by the parametric equations x(t) = 3 + t2 and y(t) = t3 + 5t.Which of the following is an
equation of the line tangent to the graph of C at the point where t = 1?
(A)
(B) y = 4x - 10
(C) y = 4x + 6
(D) y = 8x - 26
(E) y = 8x + 6

8. An object moves in the xy-plane so that its position at any time t is given by the parametric equations x(t) = t3 − 3t2
+ 2 and . What is the rate of change of y with respect to x when t = 3 ?
(A) 1/90
(B) 1/15
(C) 3/5
(D) 5/2

Answer B

This option is correct. The rate of change of y with respect to x is the derivative dy/dx.

9. If x=e2t and y=sin(2t) , then

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9.1

(A)
(B)

(C)

(D)

(E)

10. A curve C is defined by the parametric equations and . Which of the following is an
equation of the line tangent to the graph of C at the point

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

11. Consider the curve in the xy-plane represented by x=et and y=te-t for . The slope of the line tangent to the
curve at the point where x=3 is
(A) 20.086
(B) 0.342
(C) -0.005
(D) -0.011
(E) -0.033

12. A particle moves along the curve xy=10. If x=2 and dy/dt=3 , what is the value of dx/dt ?
(A) −5/2
(B) −6/5
(C) 0
(D) 4/5
(E) 6/5

13. The position of a particle moving in the xy-plane is given by the parametric functions x(t) and y(t) for which
and . What is the slope of the line tangent to the path of the particle at the point
at which t=2 ?

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9.1

(A) 0.904
(B) 1.107
(C) 1.819
(D) 2.012
(E) 3.660

14. A curve is defined by the parametric equations and , where , , , and are
nonzero constants. Which of the following gives the slope of the line tangent to the curve at the point ?
(A)

(B)

(C)
(D)

Answer B

Correct. Because and , it follows that .

15. A curve is defined by the parametric functions and , where and for all .
Which of the following equals ?

(A) 2

(B)
(C)
(D)

Answer A

Correct. Differentiating with respect to gives . It follows that

, and therefore .

16. A curve in the -plane is defined by the parametric equations and for .

Which of the following is equal to ?

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9.1

(A) 1
(B)

(C) 2
(D)

Answer C

Correct. Methods for calculating derivatives of real-valued functions can be extended to parametric
functions. Differentiating and with respect to yields and
. Substituting gives

17.

The position of a particle moving in the -plane is given by the parametric functions and , whose graphs
are shown. What is the slope of the line tangent to the path of the particle at the point at which ?
(A)

(B)

(C)
(D)

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9.1

Answer D

Correct. The slope of the line tangent to the path of the particle at the point at which is the value of
at this point. The chain rule is used to find derivatives of parametric equations.
, where the values of and are the slopes of the graphs of the linear
functions and .

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 AP CALCULUS BC Scoring Guide

9.2

1. If and , then d2y/dx2=

(A) 3/4t
(B) 3/2t
(C) 3t
(D) 6t
(E) 3/2

2. If and , what is in terms of t?

(A)

(B)
(C)
(D)
(E)

3. If x(t) = t2 + 4 and y(t) = t4 + 3, for t > 0, then in terms of t,

(A)

(B) 2
(C) 4t
(D) 6t2
(E) 12t2

4. A curve in the -plane is defined by , where and for . What is

in terms of ?
(A) 2
(B)
(C)
(D)

Answer A

Correct. The chain rule is used to find derivatives of parametric functions.

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9.2

5. A curve is defined by the parametric equations and , where and are positive constants.
What is in terms of ?
(A) 0
(B)
(C)

(D)

Answer D

Correct. The chain rule is used to find derivatives of parametric functions.

6. A curve is defined by the parametric equations and . What is in terms of ?

(A)
(B)
(C)

(D)

Answer D

Correct. The chain rule is used to find derivatives of parametric functions.

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9.2

7. A curve is defined by the parametric equations and . Which of the following is an

expression for in terms of ?

(A)

(B)
(C)
(D)

8. The position of a particle moving in the -plane is given by the parametric equations and
. At any point on the curve, the slope of the line tangent to the path of the particle is given by
ⅆ ⅆ
ⅆ
. What is the value of ⅆ at ?
(A)
(B)
(C)
(D)

Answer B

Correct. The chain rule is used to find derivatives of functions defined by parametric equations.
ⅆ ⅆ ⅆ
ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ
ⅆ ⅆ ⅆ ⅆ ⅆ
ⅆ ⅆ

The value of each derivative is computed using the calculator.

9. If and , then is

(A)

(B)

(C)

(D) undefined

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9.2

Answer C

Correct. By the chain rule, . The chain rule is needed again to find the second
derivative, as follows.

10. ⅆ
If and , what is in terms of ?
ⅆ
(A)

(B)

(C)

(D)

Answer D

Correct. The chain rule is used to find derivatives of parametric equations.

ⅆ
ⅆ ⅆ
ⅆ ⅆ
ⅆ
ⅆ ⅆ
Because ⅆ
is also given in terms of , the chain rule for parametric equations is used again to find ⅆ
.
ⅆ ⅆ ⅆ
ⅆ ⅆ ⅆ ⅆ ⅆ ⅆ

ⅆ ⅆ ⅆ ⅆ ⅆ
ⅆ ⅆ

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 AP CALCULUS BC Scoring Guide

9.3

1. The length of the curve determined by the equations and y=t from t=0 to t=4 is

(A)

(B)

(C)

(D)

(E)

2. Which of the following gives the length of the path described by the parametric equations and
from to ?

(A)

(B)

(C)

(D)

(E)

3. Which of the following gives the length of the path described by the parametric equations and
fromt=0 tot=π ?

(A)

(B)

(C)

(D)

(E)

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9.3

4. Which of the following gives the length of the curve defined by the parametric equations and
from t=0 to t=1?

(A)

(B)

(C)

(D)

(E)

5. The length of the path described by the parametric equations and , for is given
by

(A) dt

(B) dt

(C) dt

(D) dt

(E) dt

6. An ellipse is defined parametrically by and for . What is the

perimeter of the ellipse?
(A)
(B)
(C)
(D)

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9.3

Answer B

Correct. The perimeter of the ellipse is

ⅆ ⅆ .

7. What is the length of the curve defined by the parametric equations and for
?
(A) 4.221
(B) 6.511
(C) 10.819
(D) 28.267

Answer B

Correct. The length of a parametrically defined curve over the interval is given

by ⅆ . For the given curve, it follows that and

. Therefore, the length of the curve for is given by

ⅆ . This is evaluated using the graphing calculator.

8. A curve is defined by the parametric equations and , where and are constants. What is the
length of the curve from to ?
(A)
(B)

(C)

(D)

Answer C

Correct. The length of a parametrically defined curve from to is given by

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9.3

ⅆ . For the given curve, and . Therefore, the length of

the curve from to is

ⅆ ⅆ .

9. Which of the following gives the length of the curve defined by the parametric equations and
for ?

(A) ⅆ

(B) ⅆ

(C) ⅆ

(D) ⅆ

Answer B

Correct. The length of a parametrically defined curve for is given by

ⅆ . For the given curve, it follows that and

. Therefore, the length of the curve for is given by

ⅆ ⅆ.

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 AP CALCULUS BC Scoring Guide

9.6

1. For time t>0 , the position of a particle moving in the xy-plane is given by the parametric equations
and . What is the acceleration vector of the particle at time t=1 ?
(A)

(B)

(C)
(D)
(E)

2. The position of a particle moving in the xy-plane is given by the vector ⟨ ⟩, where y is a

twice-differentiable function of t. At time , what is the acceleration vector of the particle?

(A) ⟨ ⟩
(B) ⟨ ⟩
(C) ⟨ ⟩

(D) ⟨ ⟩

3. At timet≥0 , a particle moving in the xy-plane has velocity vector given by v(t) = ⟨t2, 5t⟩. What is the acceleration
vector of the particle at timet=3 ?
(A) ⟨ ⟩

(B) ⟨6, 5⟩
(C) ⟨2, 0⟩
(D)
(E)

4. For any time t≥0 , if the position of a particle in the xy-plane is given by x=t2+1 and y=ln(2t+3) , then the
acceleration vector is
(A) (2t, 2/(2t+3))
(B) (2t, −4/(2t+3)2)
(C) (2, 4/(2t+3)2)
(D) (2, 2/(2t+3)2)
(E) (2, −4/(2t+3)2)

5. A particle moves on a plane curve so that at any time t>0 its x-coordinate is t3-t and its y-coordinate is (2t-1)3. The
acceleration vector of the particle at t=1 is

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9.6

(A) (0,1)
(B) (2,3)
(C) (2,6)
(D) (6,12)
(E) (6,24)

6. For time t > 0, the position of a particle moving in the xy-plane is given by the vector ⟨ ⟩. What is the velocity
vector of the particle at time t = 2?
(A) ⟨ ⟩
(B) ⟨ ⟩
(C) ⟨ ⟩
(D) ⟨ ⟩

(E) ⟨ ⟩

7. A particle moves in the xy-plane so that at any time t its coordinates are and . At t = 1, its
acceleration vector is
(A) (0,-1)
(B) (0,12)
(C) (2, -2)
(D) (2,0)
(E) (2,8)

8. A particle moves in the xy-plane with position given by (x (t), y(t )) = (5 - 2t,t2 - 3) at time t. In which direction is
the particle moving as it passes through the point (3, -2) ?

(A) Up and to the left

(B) Down and to the left
(C) Up and to the right
(D) Down and to the right
(E) Straight up

9. The velocity vector of a particle moving in the xy-plane has components given by and
. At time t = 4, the position of the particle is (2, 1). What is the y-coordinate of the position vector at
time t = 3 ?
(A) 0.410
(B) 0.590
(C) 0.851
(D) 1.410

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9.6

10. If a particle moves in the xy-plane so that at time t>0 its position vector is , then at time
t=2, its velocity vector is

(A)

(B)
(C)
(D)
(E)

11. In the xy-plane, a particle moves along the parabola with a constant speed of units per second.
If , what is the value of when the particle is at the point ?
(A)
(B)
(C) 3
(D) 6

(E)

12. The position of a particle moving in the xy-plane is given by the parametric equations and
At which of the following points (x, y) is the particle at rest?

(A) (-4, 12)

(B) (-3, 6)
(C) (-2, 9)
(D) (0, 0)
(E) (3, 4)

13. A particle moves on the curve y=ln x so that the x-component has velocity for At time t=0,
the particle is at the point (1, 0). At time t=1, the particle is at the point
(A) (2,ln2)
(B) (e2,2)
(C)
(D) (3, ln3)
(E)

14. The position of an object moving along a path in the xy-plane is given by the parametric equations
and . The speed of the particle at time t = 0 is

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9.6

(A) 3.422
(B) 11.708
(C) 15.580
(D) 16.209

15. The position of a particle moving in the xy-plane is given by the parametric equations x(t) = cos(2t) and y(t) =
sin(2t) for time t ≥ 0. What is the speed of the particle when t = 2.3 ?
(A) 1.000
(B) 2.014
(C) 3.413
(D) 11.652

Answer C

This option is correct. The speed of the particle at time t is the magnitude of the velocity vector x'(t ),
y'(t ) at time t. When t = 2.3, the velocity vector is 3.3369, 0.7189 which has magnitude

16. For time , the position of an object moving in the xy-plane is given by the parametric equations
and . What is the speed of the object at time ?
(A) 1.155
(B) 1.319
(C) 1.339
(D) 1.810

17. The position of an object moving in the -plane is given by the parametric equations and
. At what times is the object at rest?

(A) only
(B) and only
(C) and only
(D) , , and

Answer A

Correct. The object is at rest at a time when both and .

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9.6

at and
at and .
The only time when both and is at .

18. The velocity of a particle moving in the -plane can be described by the parametric equations
and for time . If the particle is at the point at time , what is
the position of the particle at time ?
(A)
(B)
(C)
(D)

Answer B

Correct. By the Fundamental Theorem of Calculus,

19. The path of a particle in the -plane is described by the parametric equations and .
Which of the following gives the total distance traveled by the particle from to ?

(A) ⅆ

(B) ⅆ

(C) ⅆ

(D) ⅆ

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9.6

Answer B

Correct. The total distance traveled along a parametric curve from to is the definite integral

of the speed, . In this problem, and .

Therefore, the total distance traveled from to is .

20. A particle moves in the -plane so that its position at time is given by and .
What is the speed of the particle when ?
(A)
(B)
(C)
(D)

Answer D

Correct. The speed of the particle at time is given by .

The particle’s speed when is .

21. A particle moves in the -plane so that its position at any time is given by the parametric equations
and . What is the speed of the particle at time ?

(A)

(B)
(C)
(D)

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9.6

Answer C

Correct. The speed of the particle at time is .

The speed at time is therefore .

22. The position of a particle moving in the -plane is given by the parametric equations and
for time . What is the speed of the particle at time ?
(A)
(B)
(C)
(D)

Answer C

Correct. The speed of the particle at time is the magnitude of the velocity vector
at time .

23. For time , a particle moves in the -plane with velocity vector given by
. At time , the particle is at position . What is the
particle’s acceleration vector at time ?
(A)
(B)
(C)
(D)

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9.6

Answer B

Correct. The acceleration vector is the derivative of the velocity vector. The derivatives can be evaluated
numerically with the calculator.

24. For time , a particle moves with velocity vector given by , where and are continuous
functions of . At time , the particle is at position . Which of the following expressions gives the
particle’s position at time ?
(A)
(B)

(C) ⅆ ⅆ

(D) ⅆ ⅆ

Answer D

Correct. For a particle in planar motion over an interval of time, the definite integral of the velocity
vector represents the particle’s displacement over the interval of time.

ⅆ ⅆ ⅆ ⅆ

ⅆ ⅆ ⅆ ⅆ

25. A curve in the -plane is given by parametric functions and , where and
for . The coordinates of the point on the curve where are
. What is the -coordinate of the point on the curve where ?
(A)
(B) 1.654
(C) 2.878
(D) 7.410

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9.6

Answer B

Correct. The use of the Fundamental Theorem of Calculus for solving initial value problems for real-
valued functions can be extended to parametric functions. By the Fundamental Theorem of Calculus,

. Therefore,

26.
For time , the velocity of a particle moving in the -plane is given by the vector .
At time , the position of the particle is . What is the distance between the position of the particle at time
and the position of the particle at time ?
(A)
(B)
(C)
(D)

Answer A

Correct. The distance between the position at time and the position at time
is .

ⅆ ⅆ

ⅆ ⅆ

The distance is therefore .

27. The position of a particle is given by the parametric equations and . What is the
velocity vector at time ?

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9.6

(A)

(B)

(C)

(D)

Answer B

Correct. The components of the velocity vector are the derivatives of the components of the position
vector.

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9.7

1. What is the slope of the line tangent to the polar curve at ?

(A) 2
(B)
(C) 0
(D)
(E) -2

2. What is the slope of the line tangent to the polar curve when ?
(A)
(B)
(C)
(D)

3. What is the slope of the line tangent to the polar curve at the point where ?
(A)

(B)

(C)

(D)
(E)

4. What is the slope of the line tangent to the polar curve at the point ?
(A)

(B)
(C) 0
(D)
(E) 2

5. What is the slope of the line tangent to the polar curve r = 2 cos θ − 1 at the point where θ = π?
(A) -3
(B) 0
(C) 3
(D) undefined

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9.7

Answer D

This option is correct. For a polar curve given in terms of θ , the slope of the tangent line is given by

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9.7

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9.7

6. Let be a differentiable function such that and . What is the slope of the line tangent to the
polar curve at ?
(A)
(B)
(C)

(D)

Answer D
ⅆ
ⅆ ⅆ
Correct. The slope of the polar curve is ⅆ ⅆ .
ⅆ
ⅆ
ⅆ
ⅆ
ⅆ
ⅆ
ⅆ ⅆ
ⅆ ⅆ
ⅆ

7. What is the slope of the line tangent to the polar curve at the point where ?
(A)

(B)
(C)
(D)

Answer B

Correct. The slope of the polar curve is .

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9.7

8. A polar curve is given by the differentiable function for . If the line tangent to the polar
curve at is horizontal, which of the following must be true?

(A)

(B)

(C)

(D)

Answer D

Correct. For there to be a horizontal tangent line at , it must be true that at . Since
, the product rule gives
.
Note that it is not sufficient to know just that at to conclude there is a horizontal tangent
line. This is because and it is possible that at .

9. A polar curve is given by the equation for . What is the instantaneous rate of change of with
respect to when ?
(A)
(B)

(C)
(D)

Answer B

Correct. The instantaneous rate of change of with respect to can be found using the same methods as
for calculating derivatives of real-valued functions. Using the quotient rule,
. Evaluating the derivative at gives
.

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9.7

10. For a certain polar curve , it is known that and . What is

the value of at ?
(A)
(B) 0.417
(C) 1.346
(D) 3.195

Answer C

Correct. The first derivative is . The second derivative at is therefore

where the evaluations have been done with the calculator.

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10.1

1. For a series S, let , where

Which of the following statements are true?

I. S converges because the terms of S alternate and .

II. S diverges because it is not true that for all n.

III. S converges although it is not true that for all n.

(A) None
(B) I only
(C) II only
(D) III only
(E) I and III only

2.
If the series converges and for all n, which of the following must be true?

(A)

(B) for all n

(C)

(D) nan diverges

(E) converges

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10.1

3. Which of the following series diverge?

I.

II.

III.

(A) III only

(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

4.
The infinite series has nth partial sum for . What is the sum of the series ?

(A) -1
(B) 0
(C)
(D) 1
(E) The series diverges.

5.
Consider the sequence and the infinite series . Which of the following

is true?

(A) and converges.

(B) and converges.

(C) and diverges.

(D) and diverges.

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10.1

Answer C

Correct. , since the denominator goes to infinity. The partial sum of

the series, , is , and since the limit of the partial sums of

the series does not exist as a finite real number, the series diverges. This is an example of a divergent
infinite series whose terms go to 0.

6.
If , for all positive integers , what is the value of , the partial sum of the infinite series ?

(A) 1
(B)
(C)
(D)

Answer B

Correct. Each term of the infinite sequence is equal to 1. It follows that , the partial sum of

the series , is the sum of 1s. Therefore, .

7.
If for , which of the following statements about the infinite series is true?

(A) The series converges and has sum 0.

(B) The series converges and has sum .
(C) The series converges and has sum 1.
(D) The series diverges.

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10.1

Answer D

Correct. The series can be written as . The sequence of partial sums

of the series is , and this sequence of alternating 1s and 0s does not converge. Therefore,
the series diverges.

8.
The infinite series has nth partial sum for . What is the sum of the series ?

(A)

(B)
(C) 1
(D)
(E) The series diverges

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10.2

1. Which of the following series converge to 2?

I.

II.

III.

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only

2.
If , then f(1) is

(A) 0.369
(B) 0.585
(C) 2.400
(D) 2.426
(E) 3.426

3. If and are positive real numbers, which of the following conditions guarantees that the infinite series

converges?

(A)
(B)
(C)
(D)

Answer D

Correct. The given series is a geometric series with common factor and common ratio . Therefore,

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10.2

the series converges when the common ratio .

4.
is

(A)

(B)

(C)
(D) divergent

Answer B

Correct. The series is a geometric series with first term and common ratio . Since this
ratio is less than , the series converges to .

5.
Let be a real number. Which of the following statements about the infinite series is true?

(A) The sum of the series is if .

(B) The sum of the series is if .

(C) The sum of the series is if .
(D) The sum of the series is if .

Answer A

Correct. Because , the series is a geometric series with common ratio and first term 1.
The series converges if , and the sum of the series is .

6. If and are real numbers such that , which of the following infinite series has sum ?

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10.2

(A)

(B)

(C)

(D)

Answer B

Correct. For a geometric series, when . The sum of this series is

7.
What is the value of ?

(A)
(B)
(C)

(D)

(E)

8. The sum of the infinite geometric series is

(A) 1.60
(B) 2.35
(C) 2.40
(D) 2.45
(E) 2.50

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10.2

9.
What is the value of

(A) 1
(B) 2
(C) 4
(D) 6
(E) The series diverges.

10.
What is the sum of the series ?

(A)

(B)

(C)
(D)
(E) The series diverges.

11.
What is the value of ?

(A) −2
(B)

(C)
(D) 3
(E) The series diverges.

12. What is the sum of the series

(A)

(B)

(C)
(D) The series diverges.

13.
Consider the series . If a1 = 16 and for all integers n ≥ 1, then is

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10.2

(A) 0
(B) 2
(C) 17
(D) 32
(E) divergent

14.
To what number does the series converge?

(A) 0
(B)

(C)
(D) The series does not converge.

15.
Consider the geometric series where an > 0 for all n. The first term of the series is a1 = 48, and the third

term is a3 = 12. Which of the following statements about is true?

(A)

(B)

(C) converges, but the sum cannot be determined from the information given.

(D) diverges

Answer B

This option is correct. The geometric series has a common ratio of r. The sum of the series is: 48/.5 =
96.

16.
If for a constant , then

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10.2

(A)
(B)
(C)

(D)

Answer D

Correct. The series is a geometric series with first term and common ratio

. Since this ratio is less than , the series converges and

17.
If the series converges, where and , what is the sum of the series?

(A)

(B)

(C)
(D)

Answer B

Correct. This is a geometric series with ratio and first term . The series converges to
.

18.
What are all values of for which converges?

(A) only
(B) only
(C) only
(D) only

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10.2

Answer B

Correct. is a geometric series with common ratio . The series will converge

if , so if .
The only solution to is . Since , the graph of
crosses the horizontal line at and increases toward the horizontal asymptote at ,
which it never crosses since has no solution. Therefore, for all values of
satisfying .

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10.3

1. Which of the following series diverge?

I.

II.

III.

(A) II only
(B) III only
(C) I and II only
(D) I, II, and III

Answer D

Correct. Each of the series diverges because of the term test. If or does not

exist, then diverges.

does not exist.

2. The term test can be used to determine divergence for which of the following series?

I.

II.

III.

(A) III only

(B) I and III only
(C) II and III only
(D) I, II, and III

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10.3

Answer C

Correct. Series II and series III diverge by the term test. If or does not exist,

then diverges. For series II, and for series III, .

For series I, . This means that another test must be used to determine whether the
series converges or diverges. In this case, the series diverges, but the term test cannot be used to
determine divergence.

3.
If for , which of the following statements about must be true?

(A) The series converges and .

(B) The series diverges and .

(C) The series converges and .

(D) The series diverges and .

Answer D

Correct. Because , the series diverges by the term test.

4.
If and for all , which of the following must be true?

(A)

(B)

(C)

(D)

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10.3

Answer A

Correct. If , then the series would diverge by the term test. However, the series

is convergent because it is equal to . Therefore, it must be true that .

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10.5

1. Which of the following series converge?

I.

II.

III.

(A) None
(B) II only
(C) III only
(D) I and II only
(E) II and III only

2. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

3. Which of the following series converge?

I.

II.

III.

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10.5

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

4.
What are all values of p for which the infinite series converges?

(A) p > 0
(B)
(C) p > 1
(D)
(E) p > 2

5.
What are all values of p for which the series diverges?

(A)
(B) p < 1/2 only
(C)
(D) p > 1/2 only
(E) The series diverges for all p.

6. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

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10.5

7.
For what values of p will both series and converges?

(A) only
(B) only

(C) only

(D) and
(E) There are no such values of p.

8.
What are all values of p for which converges?

(A)
(B)
(C)
(D)
(E) There are no values of for which this integral converges.

9. Which of the following is a convergent -series?

(A)

(B)

(C)

(D)

Answer D

Correct. This is the -series with . Since , the series converges.

10. Which of the following is the harmonic series?

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10.5

(A)

(B)

(C)

(D)

Answer C

Correct. The harmonic series is the infinite series consisting of the reciprocals of the positive integers.

11. Which of the following is not a -series?

(A)

(B)

(C)

(D)

Answer D

Correct. This is a geometric series with common ratio .

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10.5

12. Which of the following series converge?

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

13. Which of the following series diverge?

I.

II.

III.

(A) None
(B) II only
(C) III only
(D) I and III
(E) II and III

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10.6

1.
Which of the following statements about convergence of the series is true?

(A) converges by comparison with

(B) converges by comparison with

(C) diverges by comparison with

(D) diverges by comparison with

2.
Which of the following statements about the series is true?

(A) The series diverges by the nth term test.

(B) The series diverges by limit comparison to the harmonic series .

(C) The series converges by the nth term test

(D) The series converges by limit comparison to the geometric series .

3. Which of the following series converges?

(A)

(B)

(C)

(D)

(E)

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10.6

4.
Which of the following statements about the series is true?

(A) The series diverges by the term test.

(B) The series diverges by comparison to the series .

(C) The series diverges by limit comparison to the series .

(D) The series diverges by limit comparison to the series .

Answer C

Correct. . Since the harmonic series diverges, the series

would also diverge by the limit comparison test.

5.
Which of the following statements about the series is true?

(A) The series diverges by the term test.

(B) The series diverges by comparison to the series .

(C) The series converges by comparison to the series .

(D) The series converges by comparison to the series .

Answer D

Correct. The series is a convergent geometric series. Since for

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10.6

all , the comparison test shows that the series also converges.

6. Which of the following series can be used with the limit comparison test to determine whether the series
converges or diverges?

(A)

(B)

(C)

(D)

Answer D

Correct. The limit comparison test looks at the limit of the ratio of general terms of the two positive
series. If this limit is finite and greater than 0, the two series either both converge or both diverge. For

this series, . Since the limit is finite

and nonzero and the geometric series converges, the series will also

converge by the limit comparison test.

7.
Of the following series, which can be used with the comparison test to show that diverges?

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10.6

(A)

(B)

(C)

(D)

Answer A

Correct. Since , for , for . Because the series diverges, the

comparison test can therefore be used to show that also diverges.

8. Which of the following series converge?

I.

II.

III.

(A) None
(B) I only
(C) I and II only
(D) I, II, and III

Answer B

Correct. The series converges by comparison with the convergent -series since

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10.6

for all . The series diverges by limit comparison with the harmonic

series since . The series diverges by the term

test since .

9. If for , which of the following must be true?

(A) If , then converges.

(B) If converges, then .

(C) If diverges, then .

(D) If diverges, then diverges

(E) If converges, then converges

10.
If diverges and for all n, which of the following statements must be true?

(A) converges.

(B) converges.

(C) diverges.

(D) converges.

(E) diverges.

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10.6

11. Which of the following series can be used with the limit comparison test to determine whether the series

converges or diverges?

(A)

(B)

(C)

(D)

Answer A

This option is correct. The limit comparison test looks at the limit of the ratio of general terms of the
two positive series. If this limit is finite and greater than 0, the two series either both converge or both
diverge. Since this limit is finite and nonzero, this series is suitable for use with the limit comparison test.
[Hence the series in the stem diverges because the series in (A) is the divergent harmonic series.]

12.
Consider the series and , where and for . If converges, which of the

following must be true?

(A) If , then converges.

(B) If , then diverges.

(C) If , then converges.

(D) If , then diverges.

(E) If , then the behavior of cannot be determined from the information given.

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10.7

1. The alternating series test can be used to show convergence of which of the following alternating series?

I.

II.

III.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

2. Suppose and for all . Which of the following statements must be true?

(A) diverges.

(B) converges.

(C) converges.

(D) converges.

3.
The power series has radius of convergence 2. At which of the following values of x can the

alternating series test be used with this series to verify convergence at x ?

(A) 6
(B) 4
(C) 2
(D) 0
(E) -1

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10.7

4.
Which of the following statements are true about the series , where ?

I. The series is alternating.

II. for all

III.

(A) None
(B) I only
(C) I and II only
(D) I and III only
(E) I, II, and III

5. Which of the following series converges?

(A)

(B)

(C)

(D)

Answer D

Correct. The series satisfies the three conditions: (1) the

series is alternating, (2) , and (3) the terms are decreasing since
for all . Therefore, the series converges by the alternating series test.

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10.7

6. The alternating series test can be used to show convergence for which of the following series?

I. , where
II. , where
III. ,

where

(A) I only
(B) II only
(C) I and II only
(D) I and III only

Answer A

Correct. The terms alternate in sign and decrease in absolute value, and . Therefore, the
alternating series test shows that the series in I will converge.
The terms do not alternate in sign since will take both positive and negative values as
increases. Therefore, the alternating series test does not apply to the series in II.
The terms are not decreasing in absolute value, because each term being subtracted is greater than the
previous term being added. Therefore, the alternating series test does not apply to the series in III.

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10.7

7.

Let be the function defined by . The derivative of is . The

graph of the function defined by is shown above for . Let

for all integers . Which of the following statements about the series is true?

(A) The series converges by the alternating series test.

(B) The alternating series test cannot be used to determine convergence because the series is not alternating.
(C) The alternating series test cannot be used to determine convergence because .

The alternating series test cannot be used to determine convergence because the terms are not
(D)
decreasing.

Answer D

Correct. The graph shows that for large , the values of will be both positive and negative on
intervals of approximate width . For the values of in the intervals where is positive, the
terms will increase. For the values of in the intervals where is negative, the terms will
decrease. Since this behavior will continue as grows larger, the terms in the series do not form a
decreasing sequence. Therefore, the alternating series test cannot be used.

8.
Consider the series , where for all and . Which of the following statements

is true?

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10.7

(A) The series converges conditionally.

(B) The series converges absolutely.
The series converges, but there is not enough information to determine if the series converges
(C)
absolutely.
(D) There is not enough information to determine if the series converges or diverges.

Answer D

Correct. This statement is true. Because , the term test cannot be used to conclude that
the series diverges. Because there is no information about whether the sequence is decreasing, the
alternating series test cannot be used to conclude that the series converges. No other information about
the terms of the series is provided, so no conclusion is possible about whether the series converges or
diverges.
For example, if and for , then the series converges by the alternating series test.
But if and for , and , then the series diverges because

where the series

diverges by the limit comparison test with the harmonic series.

9. Which of the following statements is true?

(A) The series diverges by the alternating series test.

(B) The series converges by the alternating series test.

(C) The series converges by the alternating series test.

(D) The series converges by the alternating series test.

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10.7

Answer D

Correct. Let . Then, the terms are positive, and .

This last inequality is true for all . Therefore, the terms of this series satisfy the conditions of the
alternating series test for , and so the series converges by that test.

10.
Which of the following statements about the series is true?

(A) The series can be shown to diverge by comparison with .

(B) The series can be shown to diverge by limit comparison with .

(C) The series can be shown to converge by comparison with .

(D) The series can be shown to converge by the alternating series test.

Answer D

Correct. The series satisfies the three conditions: (1) the series is alternating, (2)

, and (3) the terms are decreasing. Therefore, the series converges by the
alternating series test.
To verify that the terms are decreasing, consider the following.

The last inequality is true for all .

Alternatively, if , then and the function is decreasing because
for .

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10.7

11. Which of the following series converge?

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

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10.8

1. Which of the following series converges for all real numbers x ?

(A)

(B)

(C)

(D)

(E)

2. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

3.
What are all values of x for which the series converges?

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10.8

(A) All x except x = 0

(B) |x| = 3
(C)
(D) |x| > 3
(E) The series diverges for all x.

4. Which of the following series converge?

I.

II.

III.

(A) I only
(B) II only
(C) I and II only
(D) I, II, and III

Answer C

Correct. Only series I and II converge.

Series I converges by the ratio test because
.
Series II is a convergent geometric series because the common ratio is .
Series III diverges by the term test because , since the exponential term in the
numerator grows much faster than the polynomial term in the denominator as increases.

5. If for all and , which of the following series converges?

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10.8

(A)

(B)

(C)

(D)

Answer C

Correct. Since , this series would converge

by the ratio test.

6.
If the ratio test is applied to the series , which of the following inequalities results, implying that the

series converges?

(A)

(B)

(C)

(D)

Answer C

Correct. Let . The ratio test would look at the limit

. Then

because is less than 4, and therefore the series converges.

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10.8

7.
What are all positive values of for which the series will converge?

(A)
(B) only
(C) only
(D) There are no positive values of for which the series will converge.

Answer A

Correct. Let . Then for all

positive values of . By the ratio test, the series will converge for all .

8.
Consider the series . If the ratio test is applied to the series, which of the following inequalities results,

implying that the series converges?

(A)

(B)

(C)

(D)

(E)

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10.9

1. Which of the following series are conditionally convergent?

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only

Answer C

This option is correct.

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10.9

2. Which of the following series is conditionally convergent?

(A)

(B)

(C)

(D)

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10.9

Answer C

Correct. The series converges by the alternating series test:

; if , then for , showing that the

terms decrease as increases.

The series diverges, however, by the comparison test with the divergent -series

, since for all . Therefore, the series is

conditionally convergent.

3.
For what values of is the series conditionally convergent?

(A)
(B)
(C) only
(D) only

Answer C

Correct. For , or , the series is an alternating series with individual

terms that decrease in absolute value to 0. Therefore, converges for by the

alternating series test.

The series is a -series and therefore diverges for , or . Since

diverges for , the series diverges for by the limit

comparison test.

Since the series converges for and the series of absolute values

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10.9

diverges for , is conditionally convergent for .

4.
Which of the following statements is true about the series ?

(A) The series converges conditionally.

(B) The series converges absolutely.
(C) The series converges but neither conditionally nor absolutely.
(D) The series diverges.

Answer A

Correct. The series is an alternating series with individual terms that decreases in absolute

value to 0. Therefore, it converges by the alternating series test. The series of absolute values

diverges, as it is a -series with .

Therefore, is conditionally convergent.

5.
Consider the series and . Which of the following statements is true?

(A) Both series converge absolutely.

(B) Both series converge conditionally.

(C) converges absolutely, and converges conditionally.

(D) converges conditionally, and converges absolutely.

Answer D

Correct. A series is conditionally convergent if the series converges but the series of absolute

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10.9

terms diverges. It is absolutely convergent if the series converges and the series of absolute
terms also converges.
Each of the series in this problem converges by the alternating series test.

The series converges conditionally because the series diverges, since it is a

-series with .

The series converges absolutely because the series converges, since it is a -series

with .

6.
Consider the series . Which of the following statements is true?

(A) The series converges absolutely.

(B) The series converges conditionally.
(C) The series diverges.
(D) It cannot be determined whether the series converges or diverges from the information given.

Answer C

Correct. The series diverges by the term test. As , approaches .

Therefore, as , the terms are alternating between values approaching and . This

shows that does not exist. Therefore, since the terms do not approach in the limit as
, the series does not converge.

7. Which of the following series is conditionally convergent?

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10.9

(A)

(B)

(C)

(D)

8. Which of the following series are conditionally convergent?

I.

II.

III.

(A) I only
(B) II only
(C) II and III only
(D) I, II, and III

Answer C

Correct. A series is conditionally convergent if the series converges but the series of absolute
terms diverges. Each of the three series in this problem converges by the alternating series test.

The series is not conditionally convergent, since converges by the ratio test (so this

series is absolutely convergent).

The series is conditionally convergent because the series diverges, since it is a

-series with .

The series is conditionally convergent because the series diverges by the limit

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10.9

comparison test with the harmonic series .

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10.10

1.
The series converges to S. Based on the alternating series error bound, what is the least number

of terms in the series that must be summed to guarantee a partial sum that is within 0.03 of S ?
(A) 34
(B) 333
(C) 1111
(D) 9999

2.
The Taylor series for ln x, centered at x=1, is . Let f be the function given by the sum of the

first three nonzero terms of this series. The maximum value of for is
(A) 0.030
(B) 0.039
(C) 0.145
(D) 0.153
(E) 0.529

3.
The Taylor series for a function f about x = 0 is given by and converges to f for all real

numbers x. If the fourth-degree Taylor polynomial for f about x = 0 is used to approximate alternating series
error bound?
(A)
(B)

(C)

(D)

Answer C

This option is correct. Since

the fourth-degree Taylor polynomial for f is .

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10.10

Using the Taylor series for f about x = 0,

This is an alternating series and converges by the alternating series test. Therefore the alternating series
error bound can be used to approximate this value using the first two terms of the series, which is the
same as . The alternating series error bound using the first two terms in the series for is the
absolute value of the third term, , the first omitted term of the series, so
.

4.
If the infinite series is approxiately by , what is the least value of k for

which the alternating series error bound guarantees that ?

(A) 64
(B) 66
(C) 68
(D) 70

5.
Consider the series . Which of the following statements is true?

(A) The series converges, and .

(B) The series converges, and .

(C) The series converges, and .

(D) The series diverges.

Answer B

Correct. The series converges by the alternating series test. Let be the sum of the series. The sum of
the first two terms is with the next term being added. Therefore, . The alternating
series error bound guarantees that . This implies that or

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10.10

. Therefore, .

6.
If the series is approximated by the partial sum , what is the

least value

of for which the alternating series error bound guarantees that ?

(A) 31
(B) 32
(C) 33
(D) 34

Answer C

Correct. The alternating series error bound guarantees that .

Therefore, is guaranteed by the alternating series error bound if

. The least satisfying this inequality is .

7.
If the series is approximated by the partial sum with 15 terms, what is the alternating series

error bound?
(A)
(B)
(C)

(D)

Answer D

Correct. The alternating series error bound for the partial sum with 15 terms is the absolute value of the
16th term, the first omitted term of the series. This would be .

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10.10

8.
The series converges to and for all . If is the

partial sum of the series, which of the following statements must be true?
(A)

(B)
(C)
(D)

Answer D

Correct. Since the series converges, . In addition, the terms of the series are alternating and
decreasing in absolute value. Therefore, the alternating series error bound can be applied to conclude that
is bounded by , the first omitted term.

9.
The series converges to . Based on the alternating series error bound, what is the least number of

terms in the series that must be summed to guarantee a partial sum that is within of ?
(A) Two
(B) Three
(C) Four
(D) Five

Answer B

Correct. The alternating series error bound using the first terms of the series is the absolute value of the
first omitted term. The goal of this problem is to find the least such that

. The least for which this inequality holds is since and

.

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10.10

10.
The alternating series converges to . If the sum of the first six terms of the series is used to

approximate , what is the alternating series error bound?

(A)
(B)
(C)

(D)

Answer D

Correct. The alternating series error bound is the absolute value of the first omitted term in the series.
Since the first six terms of the series are used to approximate , the first omitted term corresponds to
. Therefore, the alternating series error bound is .

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10.13

1.
The power series converges conditionally at x = 5. Which of the following statements about n=0

convergence of the series at x = −4 is true?

(A) The series converges absolutely at x = −4.
(B) The series converges conditionally at x = −4.
(C) The series diverges at x = −4.
(D) There is not enough information given to determine convergence of the series at x = −4.

Answer C

This option is correct. A series is absolutely convergent on the interior of the interval of convergence
since the radius of convergence can be determined by the ratio test. Since this series is conditionally
convergent when x = 5, then 5 must be an endpoint of the interval of convergence. The series is centered
at x = 1, so the radius of convergence is 5 - 1 = 4. Thus, the series is divergent for x < -3 and x > 5.

2.
If the power series converges at and diverges at , which of the following must be

true?

I. The series converges at .

II. The series converges at .

III. The series diverges at .

(A) I only
(B) II only
(C) I and II only
(D) II and III only

3.
The interval of convergence of is

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

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4.
What are all values of x for which the series converges?

(A)
(B) only
(C) only
(D) and only
(E) and

5.
The radius of convergence for the power series is equal to 1. What is the interval of convergence?

(A) −4≤x<−2
(B) −1<x<1
(C) −1≤x<1
(D) 2<x<4
(E) 2≤x<4

6.
Which of the following is the interval of convergence for the series ?

(A) -4 < x < 0

(B)
(C) -2 < x < 0
(D)

7.
What are all values of x for which the series converges?

(A)

(B)

(C)
(D)
(E)

8.
What is the interval of convergence of the power series ?

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

9.
What are all values of x for which the series converges?

(A)
(B) -3<x<3
(C)
(D)
(E)

10.
The power series converges at . Which of the following must be true?

(A) The series diverges at .

(B) The series diverges at .
(C) The series converges at .
(D) The series converges at .
(E) The series converges at .

11.
Which of the following is the interval of convergence for the series ?

(A)
(B)
(C)
(D)

Answer A

Correct. The ratio test can be used to determine the interval of convergence.

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Now check the endpoints of the interval.

At , the series is , which converges since it is a multiple of

the alternating harmonic series.

At , the series is , which diverges since it is a multiple of the

harmonic series.
Therefore, the interval of convergence is .

12.
What are all values of for which the series converges?

(A) only
(B) only
(C) only
(D)

Answer C

Correct. The ratio test can be used to determine the interval of convergence.

Now check the endpoints of the interval:

At the series is , which converges because it is the

alternating harmonic series.

At the series is , which converges because it is the

alternating harmonic series.

Therefore the interval of convergence is .

13.
The power series converges at . Which of the following must be true?

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(A) converges at .

(B) diverges at .

(C) converges at .

(D) diverges at .

Answer C

Correct. The radius of convergence is at least , since the center is at and the series converges
at . Therefore, the interval of convergence includes the open interval . Since is in this
interval, the series must converge at .

14.

Which of the following statements about the power series above is true?
(A) The series diverges for all .
(B) The series converges for only.
(C) The series converges for only.
(D) The series converges for all .

Answer D

Correct. The ratio test can be used to determine the radius of convergence.

Therefore, the radius of convergence is infinite, and so the series converges for all .

15.
What is the radius of convergence of the series ?

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(A)
(B) 3
(C)

(D)
(E) 0

16.
The coefficients of the power series (x-2)n satisfy and for all n≥1. The radius

of convergence of the series is

(A) 0
(B)

(C)
(D) 2
(E) infinite

17. What is the radius of convergence of the Maclaurin series for ?

(A) 1/2
(B) 1
(C) 2
(D) infinite

18.
What is the radius of convergence for the power series ?

(A)
(B)

(C) 3
(D) 4
(E) 6

19.
What is the radius of convergence for the series ?

(A) 1/2
(B) 1
(C) 2
(D) 5

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Answer C

This option is correct. The ratio test can be used to determine the radius of convergence:

<1

Therefore, the radius of the convergence is 2.

20.
What is the interval of convergence of the power series ?

(A)
(B)
(C)
(D)

Answer B

Correct. is a power series about . The ratio test can be used to determine

the interior of the interval of convergence.

Evaluating the series at the endpoint yields

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. This is the harmonic series, which diverges.

Evaluating the series at the endpoint yields

. This is the alternating harmonic series, which converges.

Therefore, the interval of convergence of the power series is .

21.
Which of the following statements about the power series is true?

(A) The series does not converge for any real number .
(B) The series converges for only.
(C) The series converges on the interval only.
(D) The series converges for all real numbers .

Answer B

Correct. is a power series about . The ratio test can be used to determine the radius of

convergence of a power series.

Therefore, the radius of convergence is 0. When , the series is trivially convergent. Therefore, the
power series converges at a single point, .

22.
For what values of does the series converge?

(A) only
(B) only
(C) only
(D) The series converges for all real numbers .

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Answer D

Correct. is a power series about . The ratio test can be

used to determine the radius of convergence of a power series.

Therefore, the radius of convergence is infinite. In order words, the power series converges for all .

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