test 2 review fall 2014 sheehan calc2

1. ( points) Which of the following is a correct formula for computing the value of the series ∑ �� assuming

that −1 < � < 1, � ≠ 0, and � ≠ 0. There may be more than one correct presentation of the formula. Circle all of the correct presentations. (Alternative versions may have different equivalent formulas or a different arrangement of the answers. The actual test version will not have this many choices, I just wanted to give you a lot of examples.)

a. �

(��)

b. �

(��)�

c. ��

��

d. ��

(��) where � = 0 and � ≠ 0.

e. �

(��)�

f. ��

(��)

g. �(1 − �)��

h. (� − ��)��

2. ( points) The table below show the partial sums for the power series �� + ��(� − �)+ ��(� − �)� +��(� − �)� + ⋯ Fill in the blanks in the table with the value of the underlying terms and the value of the partial sums. (Alternative versions may have different numbers.)

Index Coefficients Value of x

Value of c

Underlying term Partial sum

0 1 3.2 3 1.000000000 1.000000000

1 0.25� 3.2 3 0.050000000 1.050000000

2 0.25� 3.2 3 0.002500000 1.052500000

3 0.25� 3.2 3 0.000125000 1.052625000

4 0.25� 3.2 3 0.000006250 1.052631250

5 0.25� 3.2 3

6 0.25� 3.2 3

3. ( points) For each series in the table below, determine if the series converges, diverges or diverges to infinity. Circle the correct answer for each series. (Alternative questions will have different numbers, different starting index, different exponents, or different bases. Make up your own and check answers with wolfram.)

Series Answer

1 + 3 + 5 + 7 + ⋯ converges, diverges , diverges to infinity

6 + 6 + 6 + 6 + ⋯ converges, diverges, diverges to infinity

5 + (−5)+ 5 + (−5)+ ⋯ converges, diverges, diverges to infinity

�1

1� + �

1

2� + �

1

3� + �

1

4� + ⋯

converges, diverges, diverges to infinity

�1

4� + �

1

4�

�

+ �1

4�

�

+ �1

4�

�

+ ⋯ converges, diverges, diverges to infinity

�1

1�

�

+ �1

2�

�

+ �1

3�

�

+ �1

4�

�


�1

4!�

��

+ �1

5!�

��

+ �1

6!�

�

+ �1

7!�

�

+ �1

8!�

�

+ �1

9!�

�


�7

2� + �

7

2�

�

+ �7

2�

�

+ �7

2�

�


� �1

4�

��

��


� �7

3�

��

��


�(−1)�

�

��


� 2�

�

��


� �1

��

��

��


� �1

��

�

��


� √2

�

��


4. ( points) Give a mathematical proof of the closed form formula for the partial sums of the terms of the geometric series. I have given you the first and last line of the proof. Assume that � ≠ 0 and � ≠ 1. (This proof starts on the bottom of page 551 and ends on the top of page 552 in the book.)

We begin by writing

�� = � + �� + �� + �� + ⋯ + ��

Since � ≠ 1, we may divide by 1 − � to obtain �� =�(�� )

(��) .

5. ( points) A ball dropped from a height of 2 meters begins to bounce. Each time it strikes the ground, it returns to 30% of its previous height. What is the total distance traveled by the ball in the first 15 bounces?

Alternative question:

The winner of a lottery receives � dollars at the end of each year for � years. The present value �� of

this prize in today’s dollars is �� = ∑ � (1 + �)�� , where � is the interest rate. Calculate �� if � =

$50,000 , � = 0.06, and � = 20.

Alternative question:

A king is building a stone wall around his garden that will be made of 75 large blocks of stone. A poor stonecutter offers to build the wall for just one grain of wheat for the first block of stone, 3 grains of wheat for the second block of stone, 9 grains of wheat for the third block of stone, 27 grains of wheat for the fourth block of stone, etc. The king agrees to this payment schedule and the stonecutter builds the wall. How many grains of wheat does the king owe the stonecutter when the wall is finished?

6. ( points) The tables below show the partial sums of the same power series for different values of �. For which

values of � does the power series seem to converge? (For the practice test I have shown different power

series. Alternative version will have different numbers.)


Value of c Partial sum

0 1 -1.02 3 1.000000000

1 0.25 -1.02 3 -0.005000000

2 0.0625 -1.02 3 1.005025000

3 0.015625 -1.02 3 -0.010050125

4 0.00390625 -1.02 3 1.010100376

5 0.000976563 -1.02 3 -0.015150878

⋮ ⋮ ⋮ ⋮ ⋮

98 9.95682E-60 -1.02 3 1.315948185

99 2.48921E-60 -1.02 3 -0.322527926

100 6.22302E-61 -1.02 3 1.324140566

101 1.55575E-61 -1.02 3 -0.330761269



0 0.25 5 3 0.250000000

1 0.0625 5 3 0.375000000

2 0.015625 5 3 0.437500000

3 0.00390625 5 3 0.468750000

4 0.000976563 5 3 0.484375000

5 0.000244141 5 3 0.492187500

98 2.48921E-60 5 3 0.500000000

99 6.22302E-61 5 3 0.500000000

100 1.55575E-61 5 3 0.500000000

101 3.88938E-62 5 3 0.500000000



0 0.25 7.1 3 0.250000000

1 0.0625 7.1 3 0.506250000

2 0.015625 7.1 3 0.768906250

3 0.00390625 7.1 3 1.038128906

4 0.000976563 7.1 3 1.314082129

5 0.000244141 7.1 3 1.596934182

98 2.48921E-60 7.1 3 105.255769279

99 6.22302E-61 7.1 3 108.137163511

100 1.55575E-61 7.1 3 111.090592598

101 3.88938E-62 7.1 3 114.117857413



0 0.25 1.5 3 0.250000000

1 0.0625 1.5 3 0.156250000

2 0.015625 1.5 3 0.191406250

3 0.00390625 1.5 3 0.178222656

4 0.000976563 1.5 3 0.183166504

5 0.000244141 1.5 3 0.181312561

98 2.48921E-60 1.5 3 0.181818182

99 6.22302E-61 1.5 3 0.181818182

100 1.55575E-61 1.5 3 0.181818182

101 3.88938E-62 1.5 3 0.181818182



0 0.25 -0.8 3 0.250000000

1 0.0625 -0.8 3 0.012500000

2 0.015625 -0.8 3 0.238125000

3 0.00390625 -0.8 3 0.023781250

4 0.000976563 -0.8 3 0.227407813

5 0.000244141 -0.8 3 0.033962578

98 2.48921E-60 -0.8 3 0.129004120

99 6.22302E-61 -0.8 3 0.127446086

100 1.55575E-61 -0.8 3 0.128926218

101 3.88938E-62 -0.8 3 0.127520093



0 0.25 -1 3 0.250000000

1 0.0625 -1 3 0.000000000

2 0.015625 -1 3 0.250000000

3 0.00390625 -1 3 0.000000000

4 0.000976563 -1 3 0.250000000

5 0.000244141 -1 3 0.000000000

98 2.48921E-60 -1 3 0.250000000

99 6.22302E-61 -1 3 0.000000000

100 1.55575E-61 -1 3 0.250000000

101 3.88938E-62 -1 3 0.000000000

(Alternate versions of this question could have different functions chosen from Table 1 pg.599 .)

(5 points) Shown below are two tables. Fill in the blank in the first column of Table 2 with the letter from Table

1 showing the Taylor series of the function centered at � = 0. (The alternate name when the series is centered

at � = 0 is Maclaurin series.)

Table 1

A.

0 + 1

1∗ �� +

−1

2∗ �� +

1

3∗ �� +

−1

4∗ �� +

1

5∗ �� +

−1

6∗ �� +

1

7∗ �� + ⋯

B.

0 + 1

1∗ �� + 0 ∗ �� +

−1

3∗ �� + 0 ∗ �� +

1

5∗ �� + 0 ∗ �� +

−1

7�� + ⋯

C.

1

0!+ 0 ∗ �� +

−1

2!∗ �� + 0 ∗ �� +

1

4!∗ �� + 0 ∗ �� +

−1

6!∗ �� + 0 ∗ �� + ⋯

D.

1 + 1 ∗ �� + 1 ∗ �� + 1 ∗ �� + 1 ∗ �� + 1 ∗ �� + 1 ∗ �� + 1 ∗ �� + ⋯

E.

0 + 1

1!∗ �� + 0 ∗ �� +

−1

3!∗ �� + 0 ∗ �� +

1

5!∗ �� + 0 ∗ �� +

−1

7!�� + ⋯

F.

1 +�

1!∗ �� +

�(� − 1)

2!∗ �� +

�(� − 1)(� − 2)

3!∗ �� +

�(� − 1)(� − 2)(� − 3)

4!∗ �� + ⋯

G.

1

0! +

1

1!∗ �� +

1

2!∗ �� +

1

3!∗ �� +

1

4!∗ �� +

1

5!∗ �� +

1

6!�� +

1

7!�� + ⋯

Table 2

Letter from table above

Function

�(�)= cos (�)

�(�)=1

(1 − �)

�(�)= ��

�(�)= tan��(�)

�(�)= ln (1 + �)

(5 points) Table 5 list consists of three columns. The first column is a list of indices. The second column lists the terms of

a sequence which underlie a series called Series A. The third column lists the terms of a sequence, which underlies

Series B. If we assume that the pattern established by the table continues, determine whether Series B converges,

diverges or remains undetermined. Circle the correct answer at the bottom of the third column.

Table 5

Index Terms of a sequence whose series is known to diverge

Terms of sequence whose convergence or divergence is to be determined

1 0.50000 0.22222

2 0.44444 0.20000

3 0.40000 0.18181

4 0.36363 0.16667

5 0.33333 0.15385

… … …

15 0.21053 0.09524

… … …

Series A (sum of terms)

Series B (sum of terms) …

Is known to diverge Converges, diverges, remains undetermined

13. (5 points) Table 6 list consists of three columns. The first column is a list of indices (that’s the word for more than

one index). The second column lists the terms of a sequence which underlie a series called Series A. The third column

lists the terms of a sequence which underlie a second series called Series B. If we assume that the pattern established by

the table continues, determine whether Series B converges, diverges or remains undetermined. Circle the correct

answer at the bottom of the third column.

Table 6

Index Terms of a sequence whose series is known to diverge

Terms of sequence whose convergence or divergence is to be determined

1 0.50000 0.70000

2 0.33333 0.46667

3 0.25000 0.35000

4 0.20000 0.28000

5 0.16666 0.23333

…

15 0.06250 0.09333

…

Series A (sum of terms)

Series B (sum of terms) …

Is known to diverge

Converges, diverges, remains undetermined

(points) Determine the first four terms of the sequence �� =�

(��)! starting with � = 1. Present your answers

first as fractions and then as decimal approximations. Your fractions should not have the factorial symbols in the answer. That means you should compute the values of the factorials. (Alternative versions may have different formulas for the terms. There were problems like this in web assign. See page 546 problems 3 and 4, 7 and 8)

Index � 1 2 3 4

Term �� as a fraction

Term �� as a decimal approx.

( points) Determine the first four terms of the recursive sequence with �� = 5 and �� = 4�� . (Alternative versions will have different numbers and may require the previous two terms for the recursion. See problem 11 page 546.)

Index � 1 2 3 4

Term ��

( points) Does the following sequence converges or diverges? (Remember that a sequence is just a list, not a

sum.)

�� =8� − 3

(4� − 5)�

If it converges, what does the limit appear to be? (Alternative versions may be similar to any problem 15-24 or

35-62 on page 546. You do not need to use the limit laws, and you will have access to your calculator.)

( points) Find a formula (general form) for the �th term of the following sequence (with a starting index of �=1.) Write the formula for the terms of the sequence in the empty box in the column labeled �. (Alternative problems will have different numbers. Some practice problems are listed below.)

Index 1 2 3 4 … �

Term ��

1

1

4

27

9

125

16

343 …

( points) Find a formula (general form) for the �th term of the following sequence (with a starting index of �=0.) Write the formula for the terms of the sequence in the empty box in the column labeled �.

Index 0 1 2 3 … �

Term ��

1

2

2

6

4

24

8

120 …

(Here are some practice problems for finding formulas for terms.)

Index 0 1 2 3 … �

Term ��

1

1

3

2

9

3

27

4 …

3�

� + 1

Index 0 1 2 3 4 … �

Term ��

1

1

5

1

25

2

125

6

625

24 …

5�

�!

Index 0 1 2 3 … �

Term ��

3

5

9

25

27

125

81

625 …

�3

5�

��

Index 0 1 2 3 … �

Term ��

3 6

4

9

16

12

64 …

3(� + 1)

4�

Index 1 2 3 4 5 … �

Term ��

1 3 6 10 15 …

�(� + 1)

2

Shown below are two tables. Table 1 gives a list of descriptions for the expressions given in Table 2. Write in the blank spaces of Table 2 the letter from the Table 1 that is associated with a correct description of the expression given in Table 2. There may be more than one correct description for some expressions. List all correct description for an expression.

Table 1

a. Geometric Sequence

b. Harmonic Sequence

c. Geometric Series

d. p-series

e. Harmonic Series

Table 2

Write the letter of all correct descriptions chosen from table above.

Expression

�� = �1

2�

�

� �

1

��

�

��

� �

1

��

��

��

� �

1

3�

��

��

� �

5

2�

��

��

�� =

1

�

( points) Given each sequence below, determine whether the limit exists for the terms of the sequence as � → ∞. If the limit is a finite value, write the value in the box in the column labeled “Limit of terms”. If the limit of the terms does not exist write DNE in the box.

For the last two sequences you are only given the general form for the �th terms. Fill in the blank boxes with decimal approximations for those terms. Use two decimal places to present the terms.

Then, choose the correct choice for the sequence by circling the correct description of the sequence.

Terms of the sequence Limit of terms

Circle one choice for each sequence

Index � 1 2 3 4 n The Sequence:

Sequence A .5

.25 .125 .0625 1

2� Converges, diverges, diverges to infinity

Sequence B 1! 2! 3! 4! �! Converges, diverges, diverges to infinity

Sequence C

1

1

1

2

1

3

1

4

1

� Converges, diverges, diverges to infinity

Sequence D �1 +�

��

� Converges, diverges, diverges to infinity

Sequence E 15� − 2

3� + 11

Converges, diverges, diverges to infinity

( points) The series � 4 ��

��

��

��= 14. Find the value of the series 6 + 7 �

�

�� + 4 �

�

��

�+ 4 �

�

��

�+ 4 �

�

��

�+ ...

( points) The series � � ��

��

��

��= 17. Find the exact value of the series 6 + 7 �

�

�� + � �

�

��

�+ � �

�

��

�+

� ��

��

�+ ...

test 2 review fall 2014 sheehan calc2

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