Deck 11: Infinite Sequences and Series

Question 1

Evaluate the indefinite integral as a power series. $\int \tan ^ { - 1 } \left( t ^ { 2 } \right) d t$

A) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 2 n + 3 } } { ( 2 n + 3 ) }$
B) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 3 } } { ( 2 n + 1 ) ( 4 n + 3 ) }$
C) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 2 n + 2 } } { ( 2 n + 1 ) }$
D) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 2 } } { ( 2 n + 1 ) ( 4 n + 3 ) }$
E) $C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 3 } } { ( 4 n + 3 ) }$

Answer 1

$C + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } t ^ { 4 n + 3 } } { ( 2 n + 1 ) ( 4 n + 3 ) }$

Question 2

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $S _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

A) $R _ { 8 } \leq 2.6130051$
B) $R _ { 8 } \geq 0.0000052$
C) $R _ { 8 } \leq 0.0000051$
D) $R _ { 8 } \geq 0.0000051$
E) $$R _ { 8 } \leq 0.000005$$

Answer 2

$R _ { 8 } \leq 0.0000051$

Question 3

Find the sum of the series.

Answer 3

Question 4

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

A) $\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { 14 ^ { n } }$
B) $\ln 14 - \sum _ { n = 1 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { t ^ { n } } { n 14 ^ { n } }$
D) $\sum _ { n = 1 } ^ { \infty } \frac { 14 t ^ { n } } { n ^ { n } }$
E) $\ln 14 + \sum _ { n = 1 } ^ { \infty } \frac { t ^ { 2 n } } { 14 ^ { n } }$

Question 5

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

A) $5 \left( 1 - 17 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
B) $5 \left( 1 - 9 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
C) $5 \left( 1 - 9 x + \frac { 115 } { 6 } x ^ { 4 } \right)$
D) $5 \left( 1 - 9 x ^ { 2 } + \frac { 97 } { 6 } x ^ { 4 } \right)$
E) $5 \left( 1 - 17 x ^ { 2 } + \frac { 67 } { 6 } x ^ { 4 } \right)$

Question 6

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

A) 0.0354
B) 0.0125
C) 0.0625
D) 0.1447
E) 0.2774

Question 7

Find the Taylor series for centered at the given value of
a. Assume that f has a power series expansion. Also find the associated radius of convergence.

Question 8

Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.

A) $1.7179$
B) $1.7189$
C) $1.7195$
D) $1.7156$
E) $1.7200$

Question 9

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

A) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$

Question 10

Evaluate the indefinite integral as an infinite series.

Question 11

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 9 + x ^ { 2 } } }$

A) $| x | < 1$
B) $| x | < 10$
C) $| x | < 6$
D) $| x | < 9$
E) $| x | < 3$

Question 12

Find the Maclaurin series for f and its radius of convergence.

Question 13

Use the binomial series to expand the function as a power series. Find the radius of convergence.

Question 14

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

A) 118.933
B) 114.133
C) 34.3233
D) 114.333
E) 115.933

Question 15

Find a power series representation for the indefinite integral.

Question 16

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$

A) $\sum _ { n = 0 } ^ { \infty } 22 y ^ { 2 n + 1 }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { 2 y ^ { 2 n + 1 } } { 11 ^ { n + 1 } ( 2 n + 1 ) }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { y ^ { 2 n + 1 } } { 11 ^ { n + 1 } ( n + 1 ) }$
D) $\sum _ { n = 7 } ^ { \infty } \frac { 22 y ^ { 2 n + 1 } } { 22 }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { y ^ { 2 n + 1 } } { 11 }$

Question 17

Find the Maclaurin series for using the definition of a Maclaurin serires.

Question 18

Find a power series representation for the function and determine the radius of convergence.

Question 19

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .

A) $0.7074$
B) $4.2074$
C) $3.2074$
D) $2.2074$
E) $1.2074$

Question 20

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$

A) $R = 7 , I = ( - 7,7 )$
B) $R = 0 , I = \{ 0 \}$
C) $R = 7 , I = [ - 7,7 ]$
D) $R = \infty , I = ( - \infty , \infty )$

Question 21

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 2 } ^ { \infty } \frac { x ^ { n } } { n ( \ln n ) ^ { 8 } }$

A) $R = 0 , I = \{ 0 \}$
B) $R = 1 , I = [ - 1,1 ]$
C) $R = 1 , I = ( - 1,1 )$
D) $R = \infty , I = ( - \infty , \infty )$

Question 22

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$

A) $R = 0 , I = \{ 0 \}$
B) $R = \infty , I = ( - \infty , \infty )$
C) $R = 6 , I = [ - 6,6 ]$
D) $R = 6 , I = ( - 6,6 )$

Question 23

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is $9$ . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

A) $252$
B) $3$
C) $1$
D) $256$
E) $16$

Question 24

Find the interval of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }$

A) $[ - 1,1 ]$
B) $( - 1,1 )$
C) $( - 1,1 ]$
D) diverges everywhere
E) $[ - 1,1 )$

Question 25

Find the interval of convergence of the series.

Question 26

Test the series for convergence or divergence.

Question 27

Find the radius of convergence and the interval of convergence of the power series.

Question 28

Find the radius of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 3 } x ^ { n } } { 2 ^ { n } }$

A) $R = 0$
B) $R = \infty$
C) $R = 1$
D) $R = \frac { 1 } { 2 }$
E) $R = 2$

Question 29

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

A) $R = 1 , I = [ 7,9 )$
B) $R = 1 , I = ( 7,9 ]$
C) $R = 8 , I = [ - 8,8 )$
D) $R = 8 , I = ( - 8,8 )$

Question 30

Test the series for convergence or divergence.

Question 31

Determine whether the series is convergent or divergent.

Question 32

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { 3 \cdot 6 \cdot 9 \cdot \cdots \cdot 3 n } { 4 \cdot 7 \cdot 10 \cdot \cdots \cdot ( 3 n + 1 ) } x ^ { 2 n + 1 }$

A) $R = \infty , I = ( - \infty , \infty )$
B) $R = 1 , I = ( - 1,1 )$
C) $R = 0 , I = \{ 0 \}$
D) $R = 1 , I = [ - 1,1 ]$

Question 33

Find the radius of convergence of the series. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { ( x + 10 ) ^ { n } } { n 6 ^ { n } }$

A) $( - 8,6 ]$
B) $( 2,14 ]$
C) $( - 14 , - 2 )$
D) $[ - 16 , - 4 )$
E) $[ - 1,1 ]$

Question 34

Find the radius of convergence and the interval of convergence of the power series.

Question 35

Determine whether the series converges or diverges.

Question 36

Test the series for convergence or divergence.

Question 37

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } ( x - 6 ) ^ { n } } { n \cdot 5 ^ { n } }$

A) $R = 6 , I = ( - 1,11 ]$
B) $R = 5 , I = ( 1,11 ]$
C) $R = \frac { 1 } { 5 } , I = \left[ - \frac { 1 } { 5 } , \frac { 1 } { 5 } \right)$
D) $R = 5 , I = [ - 5,5 )$

Question 38

Use the binomial series to expand the function as a power series. Find the radius of convergence.

Question 39

Determine whether the series is convergent or divergent.

Question 40

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { n + 2 }$

A) $R = 2 , I = [ - 2,2 )$
B) $R = 1 , I = ( - 1,1 )$
C) $R = 1 , I = [ - 1,1 )$
D) $R = 2 , I = ( - 2,2 )$

Question 41

Which of the given series are absolutely convergent?

A) $\sum _ { n = 1 } ^ { \infty } \frac { \cos \frac { \pi n } { 8 } } { n \sqrt { n } }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { \sin 5 n } { n }$

Question 42

Determine whether the series is convergent or divergent.

Question 43

Determine whether the series converges or diverges.

Question 44

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }$

A) absolutely convergent
B) divergent
C) conditionally convergent

Question 45

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

A) absolutely convergent
B) divergent
C) conditionally convergent

Question 46

Test the series for convergence or divergence.

Question 47

Use the sum of the first 9 terms to approximate the sum of the following series. Write your answer to six decimal places.

Question 48

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Question 49

Determine whether the series converges or diverges.

Question 50

Find the partial sum $S _ { 7 }$ of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 + 4 ^ { n } }$ . Give your answer to five decimal places.

A) $S _ { 7 } = 0.18975$
B) $S _ { 7 } = 0.19176$
C) $S _ { 7 } = 0.18985$
D) $S _ { 7 } = 0.21555$
E) $S _ { 7 } = 1.60976$

Question 51

Determine whether the series is convergent or divergent.

Question 52

Determine whether the series converges or diverges.

Question 53

Test the series for convergence or divergence.

Question 54

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Question 55

For which positive integers k is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( k n ! ) }$ convergent?

A) $k \geq 4$
B) $k \leq - 4$
C) $k \leq 0$
D) $k \geq 0$
E) $k \geq 1$

Question 56

Determine whether the series converges or diverges.

Question 57

Determine whether the series converges or diverges.

Question 58

Determine whether the series converges or diverges.

Question 59

Determine whether the series is convergent or divergent.

Question 60

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Question 61

Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

A) $6.97036$
B) $4.97036$
C) $7.97036$
D) $3.97036$
E) $5.97036$

Question 62

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0399$

A) $n = 6$
B) $n = 8$
C) $n = 5$
D) $n = 12$
E) $n = 13$

Question 63

Determine the number of terms sufficient to obtain the sum of the series accurate to three decimal places. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 5 ) ^ { n + 3 } } { ( n + 1 ) ! }$

A) 19
B) 22
C) 20
D) 21

Question 64

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

A) The series is convergent.
B) The series is divergent.

Question 65

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }$

A) -0.95
B) -1.02
C) -0.87
D) -1.09

Question 66

Approximate the sum to the indicated accuracy.

Question 67

Which of the partial sums of the alternating series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n }$ are overestimates of the total sum?

A) $S _ { 88 }$
B) $S _ { 98 }$
C) $s _ { 65 }$
D) $S _ { 77 }$
E) $S _ { 89 }$

Question 68

Test the series for convergence or divergence.

Question 69

Let $a _ { k } = f ( k )$ where f is a continuous, positive, and decreasing function on $[ n , \infty ) ,$ and suppose that $\sum _ { k = 1 } ^ { \infty } a _ { k }$ is convergent. Defining $R _ { n } = S - S _ { n } ,$ where $S = \sum _ { n = 1 } ^ { \infty } a _ { n }$ and $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ we have that $\int _ { n + 1 } ^ { \infty } f ( x ) d x \leq R _ { n } \leq \int _ { n } ^ { \infty } f ( x ) d x$ Find the maximum error if the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 8 } { n ^ { 2 } }$ is approximated by $S _ { 40 }$

A) 0.2
B) 0.025
C) 0.0006
D) 0.005

Question 70

Use the Comparison Test to determine whether the series is convergent or divergent.

Question 71

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Write a such that .

Question 72

Determine whether the series is convergent or divergent.

Question 73

Test the series for convergence or divergence.

Question 74

Test the series for convergence or divergence.

Question 75

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 2 } + 5 }$

A) converges
B) diverges

Question 76

Determine whether the series is convergent or divergent.

Question 77

Determine which series is convergent.

A) $- \frac { 2 } { 7 } + \frac { 3 } { 8 } - \frac { 4 } { 9 } + \frac { 5 } { 10 } - \frac { 6 } { 11 } - \ldots$
B) $\frac { 4 } { 3 } - \frac { 4 } { 4 } + \frac { 4 } { 5 } - \frac { 4 } { 6 } + \frac { 4 } { 7 } -$

Question 78

Approximate the sum to the indicated accuracy.

Question 79

Determine whether the series is convergent or divergent.

Question 80

Determine whether the series is convergent or divergent.