Question 1

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence.
-$a_{1}=2, a_{n+1}=(-1)^{n} a_{n}$

Accepted Answer

A)  2, -2, -2, 2, 2 
B)  2, -2, 2, -2, 2 
C)  -2, 2, -2, 2, -2 
D)  2, 2, -2, -2, 2 
A)  2, -2, -2, 2, 2 
B)  2, -2, 2, -2, 2 
C)  -2, 2, -2, 2, -2 
D)  2, 2, -2, -2, 2 A

Question 2

Find the first four terms of the binomial series for the given function.
-$$\left( 1 - \frac { x } { 7 } \right) ^ { 1 / 3 }$$

Accepted Answer

A)  $1 - \frac { 1 } { 21 } x - \frac { 1 } { 147 } x ^ { 2 } - \frac { 5 } { 9261 } x ^ { 3 }$ 
B)  $1 - \frac { 1 } { 21 } x + \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 9261 } x ^ { 3 }$ 
C)  $1 - \frac { 1 } { 21 } x - \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 27,783 } x ^ { 3 }$ 
D)  $1 - \frac { 1 } { 21 } x + \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 27,783 } x ^ { 3 }$ 
A)  $1 - \frac { 1 } { 21 } x - \frac { 1 } { 147 } x ^ { 2 } - \frac { 5 } { 9261 } x ^ { 3 }$ 
B)  $1 - \frac { 1 } { 21 } x + \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 9261 } x ^ { 3 }$ 
C)  $1 - \frac { 1 } { 21 } x - \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 27,783 } x ^ { 3 }$ 
D)  $1 - \frac { 1 } { 21 } x + \frac { 1 } { 441 } x ^ { 2 } - \frac { 5 } { 27,783 } x ^ { 3 }$ C

Question 3

Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n + 1 } } { 10 ^ { n - 1 } }$$

Accepted Answer

A)  Converges; $\frac { 700 } { 3 }$ 
B)  Converges; $\frac { 490 } { 3 }$ 
C)  Converges; $\frac { 70 } { 3 }$ 
D)  Diverges 
A)  Converges; $\frac { 700 } { 3 }$ 
B)  Converges; $\frac { 490 } { 3 }$ 
C)  Converges; $\frac { 70 } { 3 }$ 
D)  Diverges B

Question 4

Provide an appropriate response.
-If $\sin x$ is replaced by $x - \frac { x ^ { 3 } } { 6 }$ and $| x | < 0.5$, what estimate can be made of the error?

Accepted Answer

A)  $| \mathrm { E } | < 0.00052083$ 
B)  $| \mathrm { E } | < 0.00026042$ 
C)  $| \mathrm { E } | < 0.0026042$ 
D)  $| E | < 0.0013021$ 
A)  $| \mathrm { E } | < 0.00052083$ 
B)  $| \mathrm { E } | < 0.00026042$ 
C)  $| \mathrm { E } | < 0.0026042$ 
D)  $| E | < 0.0013021$

Question 5

Use the limit comparison test to determine if the series converges or diverges.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 3 } { 9 + 8 n ( \ln n ) ^ { 2 } }$$

Accepted Answer

A)  Converges 
B)  Diverges 
A)  Converges 
B)  Diverges

Question 6

Find the infinite sum accurate to three decimal places.
-$$\sum _ { n = 1 } ^ { \infty } \frac { 2 ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }$$

Accepted Answer

A)  1.500 
B)  3.000 
C)  0.500 
D)  1.000 
A)  1.500 
B)  3.000 
C)  0.500 
D)  1.000

Question 7

Find the series' radius of convergence.
-$$\sum _ { n = 2 } ^ { \infty } \frac { x ^ { 4 n } } { ( \ln n ) ^ { 4 } }$$

Accepted Answer

A)  $\infty$, for all $x$ 
B)  1 
C)  0 
D)  2 
A)  $\infty$, for all $x$ 
B)  1 
C)  0 
D)  2

Question 8

Solve the problem.
-A ball is dropped from a height of $21 \mathrm {~m}$ and always rebounds $\frac { 1 } { 3 }$ of the height of the previous drop. How far does it travel (up and down) before coming to rest?

Accepted Answer

A)  42 m 
B)  10.5 m 
C)  63 m 
D)  31.5 m 
A)  42 m 
B)  10.5 m 
C)  63 m 
D)  31.5 m

Question 9

Find the first four terms of the binomial series for the given function.
-$\left( 1 - 8 x ^ { 2 } \right) ^ { - 1 / 2 }$

Accepted Answer

A)  $1 + 4 x ^ { 2 } + 24 x ^ { 4 } + 160 x ^ { 6 }$ 
B)  $1 + 4 x ^ { 2 } + 48 x ^ { 4 } + 576 x ^ { 6 }$ 
C)  $1 + 4 x ^ { 2 } + 24 x ^ { 4 } + 288 x ^ { 6 }$ 
D)  $1 + 4 x ^ { 2 } + 48 x ^ { 4 } + 320 x ^ { 6 }$ 
A)  $1 + 4 x ^ { 2 } + 24 x ^ { 4 } + 160 x ^ { 6 }$ 
B)  $1 + 4 x ^ { 2 } + 48 x ^ { 4 } + 576 x ^ { 6 }$ 
C)  $1 + 4 x ^ { 2 } + 24 x ^ { 4 } + 288 x ^ { 6 }$ 
D)  $1 + 4 x ^ { 2 } + 48 x ^ { 4 } + 320 x ^ { 6 }$

Question 10

$\text { Determine if the series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { defined by the formula converges or diverges. }$
-$$\mathrm { a } _ { 1 } = \frac { 1 } { 9 } , \mathrm { a } _ { \mathrm { n } + 1 } = \sqrt [ n ] { \mathrm { a } _ { \mathrm { n } } }$$

Accepted Answer

A)  Diverges 
B)  Converges 
A)  Diverges 
B)  Converges

Question 11

Provide an appropriate response.
-For approximately what values of $x$ can $\sin x$ be replaced by $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$ with an error of magnitude no greater than $5 \times 10 ^ { - 6 } ?$

Accepted Answer

A)  $| x | < 0.59106$ 
B)  $| x | < 0.45942$ 
C)  $| x | < 0.54146$ 
D)  $| x | < 0.40357$ 
A)  $| x | < 0.59106$ 
B)  $| x | < 0.45942$ 
C)  $| x | < 0.54146$ 
D)  $| x | < 0.40357$

Question 12

By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$; otherwise indicate divergence.
-$$a _ { n } = \cos n$$

Accepted Answer

A)  L = 1, N = 394 
B)  L = 0, N = 237 
C)  L = 0, N = 394 
D)  diverges 
A)  L = 1, N = 394 
B)  L = 0, N = 237 
C)  L = 0, N = 394 
D)  diverges

Question 13

Solve the problem.
-$\text { If } \sum a _ { n } \text { is a convergent series of nonnegative terms, what can be said about } \sum \text { na } { } _ { n } \text { ? }$

Accepted Answer

A)  Always converges 
B)  May converge or diverge 
C)  Always diverges 
A)  Always converges 
B)  May converge or diverge 
C)  Always diverges

Question 14

Find the interval of convergence of the series.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( x - 8 ) ^ { n } } { \ln ( n + 8 ) }$$

Accepted Answer

A)  $7 < x < 9$ 
B)  $- \infty < x < \infty$ 
C)  $x < 9$ 
D)  $7 \leq x < 9$ 
A)  $7 < x < 9$ 
B)  $- \infty < x < \infty$ 
C)  $x < 9$ 
D)  $7 \leq x < 9$

Question 15

Find the first four terms of the binomial series for the given function.
-$$( 1 + 4 x ) ^ { - 1 / 2 }$$

Accepted Answer

A)  $1 - 2 x + 6 x ^ { 2 } - 20 x ^ { 3 }$ 
B)  $1 - 2 x - 12 x ^ { 2 } - 20 x ^ { 3 }$ 
C)  $1 - 2 x + 6 x ^ { 2 } - 30 x ^ { 3 }$ 
D)  $1 - 2 x + 6 x ^ { 2 } - 10 x ^ { 3 }$ 
A)  $1 - 2 x + 6 x ^ { 2 } - 20 x ^ { 3 }$ 
B)  $1 - 2 x - 12 x ^ { 2 } - 20 x ^ { 3 }$ 
C)  $1 - 2 x + 6 x ^ { 2 } - 30 x ^ { 3 }$ 
D)  $1 - 2 x + 6 x ^ { 2 } - 10 x ^ { 3 }$

Question 16

Find the limit of the sequence if it converges; otherwise indicate divergence.
-$$a _ { n } = \sqrt [ n ] { 5 ^ { n } \cdot n }$$

Accepted Answer

A)  1 
B)  5 
C)  0 
D)  Diverges 
A)  1 
B)  5 
C)  0 
D)  Diverges

Question 17

Find the Maclaurin series for the given function.
-$e ^ { 4 x }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n ! }$ 
B)  $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } 4 ^ { \mathrm { n } } \mathrm { x } ^ { \mathrm { n } } } { \mathrm { n } ! }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { n } x ^ { n } } { n ! }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n ! }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n ! }$ 
B)  $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } 4 ^ { \mathrm { n } } \mathrm { x } ^ { \mathrm { n } } } { \mathrm { n } ! }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { n } x ^ { n } } { n ! }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } x ^ { n } } { n ! }$

Question 18

Express the number as the ratio of two integers.
-0.8333333 . . .

Accepted Answer

A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 25 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 5 } { 2 }$ 
A)  $\frac { 5 } { 6 }$ 
B)  $\frac { 25 } { 3 }$ 
C)  $\frac { 5 } { 3 }$ 
D)  $\frac { 5 } { 2 }$

Question 19

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$\frac { x ^ { 7 } } { 1 - 3 x }$$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 3 ^ { n } n _ { n + 7 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } 3 ^ { n _ { x } n + 8 }$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n _ { 3 } } n _ { x } n + 8$ 
D)  $\sum _ { n = 0 } ^ { \infty } 3 ^ { n _ { x } n + 7 }$ 
A)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 3 ^ { n } n _ { n + 7 }$ 
B)  $\sum _ { n = 0 } ^ { \infty } 3 ^ { n _ { x } n + 8 }$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n _ { 3 } } n _ { x } n + 8$ 
D)  $\sum _ { n = 0 } ^ { \infty } 3 ^ { n _ { x } n + 7 }$

Question 20

Find the values of x for which the geometric series converges.
-$$\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$$

Accepted Answer

A)  $\{ x \mid x$ is not a multiple of $2 \pi \}$ 
B)  $\{ x \mid x$ is not a multiple of $\pi \}$ 
C)  $- \infty < x < \infty$ 
D)  diverges for all $x$ 
A)  $\{ x \mid x$ is not a multiple of $2 \pi \}$ 
B)  $\{ x \mid x$ is not a multiple of $\pi \}$ 
C)  $- \infty < x < \infty$ 
D)  diverges for all $x$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a_{1}=2, a_{n+1}=(-1)^{n} a_{n}$

Find the first four terms of the binomial series for the given function. - $\left( 1 - \frac { x } { 7 } \right) ^ { 1 / 3 }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n + 1 } } { 10 ^ { n - 1 } }$

Provide an appropriate response. -If $\sin x$ is replaced by $x - \frac { x ^ { 3 } } { 6 }$ and $| x | < 0.5$ , what estimate can be made of the error?

Use the limit comparison test to determine if the series converges or diverges. - $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { 9 + 8 n ( \ln n ) ^ { 2 } }$

Find the infinite sum accurate to three decimal places. - $\sum _ { n = 1 } ^ { \infty } \frac { 2 ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }$

Find the series' radius of convergence. - $\sum _ { n = 2 } ^ { \infty } \frac { x ^ { 4 n } } { ( \ln n ) ^ { 4 } }$

Solve the problem. -A ball is dropped from a height of $21 \mathrm {~m}$ and always rebounds $\frac { 1 } { 3 }$ of the height of the previous drop. How far does it travel (up and down) before coming to rest?

Find the first four terms of the binomial series for the given function. - $\left( 1 - 8 x ^ { 2 } \right) ^ { - 1 / 2 }$

$\text { Determine if the series } \sum _ { n = 1 } ^ { \infty } a _ { n } \text { defined by the formula converges or diverges. }$ - $\mathrm { a } _ { 1 } = \frac { 1 } { 9 } , \mathrm { a } _ { \mathrm { n } + 1 } = \sqrt [ n ] { \mathrm { a } _ { \mathrm { n } } }$

Provide an appropriate response. -For approximately what values of $x$ can $\sin x$ be replaced by $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$ with an error of magnitude no greater than $5 \times 10 ^ { - 6 } ?$

By calculating an appropriate number of terms, determine if the series converges or diverges. If it converges, find the limit $L$ and the smallest integer $N$ such that $\left| a _ { n } - L \right| < 0.01$ for $n \geq N$ ; otherwise indicate divergence. - $a _ { n } = \cos n$

Solve the problem. - $\text { If } \sum a _ { n } \text { is a convergent series of nonnegative terms, what can be said about } \sum \text { na } { } _ { n } \text { ? }$

Find the interval of convergence of the series. - $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 8 ) ^ { n } } { \ln ( n + 8 ) }$

Find the first four terms of the binomial series for the given function. - $( 1 + 4 x ) ^ { - 1 / 2 }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \sqrt [ n ] { 5 ^ { n } \cdot n }$

Find the Maclaurin series for the given function. - $e ^ { 4 x }$

Express the number as the ratio of two integers. -0.8333333 . . .

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { x ^ { 7 } } { 1 - 3 x }$

Find the values of x for which the geometric series converges. - $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$

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Exam 11: Infinite Sequences and Series

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - a1=2,an+1=(−1)nana_{1}=2, a_{n+1}=(-1)^{n} a_{n}a1​=2,an+1​=(−1)nan​

Find the first four terms of the binomial series for the given function. - (1−x7)1/3\left( 1 - \frac { x } { 7 } \right) ^ { 1 / 3 }(1−7x​)1/3

Determine if the series converges or diverges; if the series converges, find its sum. - ∑n=1∞7n+110n−1\sum _ { n = 1 } ^ { \infty } \frac { 7 ^ { n + 1 } } { 10 ^ { n - 1 } }∑n=1∞​10n−17n+1​

Provide an appropriate response. -If sin⁡x\sin xsinx is replaced by x−x36x - \frac { x ^ { 3 } } { 6 }x−6x3​ and ∣x∣<0.5| x | < 0.5∣x∣<0.5 , what estimate can be made of the error?

Use the limit comparison test to determine if the series converges or diverges. - ∑n=1∞39+8n(ln⁡n)2\sum _ { n = 1 } ^ { \infty } \frac { 3 } { 9 + 8 n ( \ln n ) ^ { 2 } }∑n=1∞​9+8n(lnn)23​

Find the infinite sum accurate to three decimal places. - ∑n=1∞2(−1)n+13n\sum _ { n = 1 } ^ { \infty } \frac { 2 ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }∑n=1∞​3n2(−1)n+1​

Find the series' radius of convergence. - ∑n=2∞x4n(ln⁡n)4\sum _ { n = 2 } ^ { \infty } \frac { x ^ { 4 n } } { ( \ln n ) ^ { 4 } }∑n=2∞​(lnn)4x4n​

Solve the problem. -A ball is dropped from a height of 21 m21 \mathrm {~m}21 m and always rebounds 13\frac { 1 } { 3 }31​ of the height of the previous drop. How far does it travel (up and down) before coming to rest?

Find the first four terms of the binomial series for the given function. - (1−8x2)−1/2\left( 1 - 8 x ^ { 2 } \right) ^ { - 1 / 2 }(1−8x2)−1/2

Provide an appropriate response. -For approximately what values of xxx can sin⁡x\sin xsinx be replaced by x−x36+x5120x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }x−6x3​+120x5​ with an error of magnitude no greater than 5×10−6?5 \times 10 ^ { - 6 } ?5×10−6?

Find the interval of convergence of the series. - ∑n=1∞(x−8)nln⁡(n+8)\sum _ { n = 1 } ^ { \infty } \frac { ( x - 8 ) ^ { n } } { \ln ( n + 8 ) }∑n=1∞​ln(n+8)(x−8)n​

Find the first four terms of the binomial series for the given function. - (1+4x)−1/2( 1 + 4 x ) ^ { - 1 / 2 }(1+4x)−1/2

Find the limit of the sequence if it converges; otherwise indicate divergence. - an=5n⋅nna _ { n } = \sqrt [ n ] { 5 ^ { n } \cdot n }an​=n5n⋅n​

Find the Maclaurin series for the given function. - e4xe ^ { 4 x }e4x