Question 1

Express the given decimal as a fraction. $1.441441441 \ldots$

Accepted Answer

A)  $\frac { 160 } { 111 }$ 
B)  $\frac { 174 } { 143 }$ 
C)  $\frac { 49 } { 111 }$ 
D)  $\frac { 60 } { 11 }$ 
A)  $\frac { 160 } { 111 }$ 
B)  $\frac { 174 } { 143 }$ 
C)  $\frac { 49 } { 111 }$ 
D)  $\frac { 60 } { 11 }$ A

Question 2

Use summation notation to write the given series in compact form. $$\frac { 1 } { 4 } - \frac { 2 } { 16 } + \frac { 3 } { 64 } - \frac { 4 } { 256 } + \cdots$$

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } n ( - 1 ) ^ { n - 1 } \left( \frac { 1 } { 4 } \right) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { - n } { 4 ^ { n } }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { n } { 4 ^ { n } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \left( - \frac { n } { 4 } \right) ^ { n }$ 
A)  $\sum _ { n = 1 } ^ { \infty } n ( - 1 ) ^ { n - 1 } \left( \frac { 1 } { 4 } \right) ^ { n }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { - n } { 4 ^ { n } }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { n } { 4 ^ { n } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \left( - \frac { n } { 4 } \right) ^ { n }$ A

Question 3

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k - 9 } { k ^ { 2 } + 6 }$

Accepted Answer

A) True 
 B)False  True

Question 4

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 4 } { 6 ^ { k } + k }$

Accepted Answer

A) True 
 B)False

Question 5

Use a Taylor polynomial of specified degree n to approximate the indicated quantity.Round to four decimal places. $\sqrt { 361 } ; n = 3$

Accepted Answer

A) 6.0071 
B) 6.0083 
C) 6.0715 
D) 6.0166 
A) 6.0071 
B) 6.0083 
C) 6.0715 
D) 6.0166

Question 6

Find the fifth partial sum S5 of the given series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$

Accepted Answer

A)  $- \frac { 121 } { 243 }$ 
B)  $- \frac { 40 } { 81 }$ 
C)  $- \frac { 61 } { 243 }$ 
D)  $- \frac { 20 } { 81 }$ 
A)  $- \frac { 121 } { 243 }$ 
B)  $- \frac { 40 } { 81 }$ 
C)  $- \frac { 61 } { 243 }$ 
D)  $- \frac { 20 } { 81 }$

Question 7

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }$

Accepted Answer

A) Converges to $- \frac { 5 } { 14 }$ 
B) Converges to $\frac { 5 } { 4 }$ 
C) Diverges 
D) Converges to $\frac { 9 } { 14 }$ 
A) Converges to $- \frac { 5 } { 14 }$ 
B) Converges to $\frac { 5 } { 4 }$ 
C) Diverges 
D) Converges to $\frac { 9 } { 14 }$

Question 8

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$

Accepted Answer

A)  $R =$ 4 
B)  $R =$ $\infty$ 
C)  $R = \frac { 1 } { 4 }$ 
D)  $R =$ 0 
A)  $R =$ 4 
B)  $R =$ $\infty$ 
C)  $R = \frac { 1 } { 4 }$ 
D)  $R =$ 0

Question 9

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 3 } } { 2 ^ { k } }$

Accepted Answer

A) True 
 B)False

Question 10

If  $\sum _ { k = 1 } ^ { \infty } a _ { k } = - 4 \text { and } \sum _ { k = 1 } ^ { \infty } b _ { k } = 9 \text {, find } \sum _ { k = 1 } ^ { \infty } \left( 9 a _ { k } - 2 b _ { k } \right)$

Accepted Answer

A) 18 
B) -54 
C) -13 
D) -18 
A) 18 
B) -54 
C) -13 
D) -18

Question 11

Find a power series for the given function. $f ( x ) = \frac { x } { 2 - x }$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n + 1 } }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n } }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 2 } \right) ^ { n }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n + 1 } }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n + 1 } } { 2 ^ { n } }$

Question 12

Suppose that nationwide,approximately 91% of all income is spent and 9% is saved.What is the total amount of spending generated by a 55 billion dollar tax rebate if savings habits do not change?

Accepted Answer

A) $611 billion 
B) $105 billion 
C) $556 billion 
D) $60 billion 
A) $611 billion 
B) $105 billion 
C) $556 billion 
D) $60 billion

Question 13

Determine the interval of absolute convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } k ^ { 2 } \left( \frac { x } { 6 } \right) ^ { k }$

Accepted Answer

A)  $- 6 < x < 6$ 
B) All real x 
C)  $- 1 < x < 1$ 
D)  $- \frac { 1 } { 6 } < x < \frac { 1 } { 6 }$ 
A)  $- 6 < x < 6$ 
B) All real x 
C)  $- 1 < x < 1$ 
D)  $- \frac { 1 } { 6 } < x < \frac { 1 } { 6 }$

Question 14

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k ! x ^ { k } } { 6 ^ { k } }$

Accepted Answer

A)  $R =$ $\frac { 1 } { 6 }$ 
B)  $R =$ 6 
C)  $R =$ $\infty$ 
D)  $R =$ 0 
A)  $R =$ $\frac { 1 } { 6 }$ 
B)  $R =$ 6 
C)  $R =$ $\infty$ 
D)  $R =$ 0

Question 15

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = e ^ { 2 x } + e ^ { - 2 x }$ ; a = 0

Accepted Answer

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

Question 16

the given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 9 ^ { k } }$

Accepted Answer

A) True 
 B)False

Question 17

Use a Taylor polynomial of specified degree n together with term-by-term integration to estimate the indicated definite integral.Round to six decimal places $\int _ { 0 } ^ { 0.4 } e ^ { - x ^ { 2 } } d x , n = 6$

Accepted Answer

A) 0.422396 
B) 1.173483 
C) 0.469393 
D) 0.379652 
A) 0.422396 
B) 1.173483 
C) 0.469393 
D) 0.379652

Question 18

A patient is given an injection of 21 units of a certain drug every 24 hours.The drug is eliminated exponentially so that the fraction that remains in the patient's body after t days is $f ( t ) = e ^ { - t / 5 }$ If the treatment is continued indefinitely,approximately how many units of the drug will eventually be in the patient's body just prior to an injection?

Accepted Answer

A) 94.85 units 
B) 46.65 units 
C) 115.85 units 
D) 141.50 units 
A) 94.85 units 
B) 46.65 units 
C) 115.85 units 
D) 141.50 units

Question 19

Find the Taylor series about $x = 0$ for the indefinite integral $\int \frac { x } { 1 - 5 x ^ { 2 } } d x$

Accepted Answer

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

Question 20

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 3 } { 7 ^ { n } }$

Accepted Answer

A) Converges to $\frac { 21 } { 8 }$ 
B) Diverges. 
C) Converges to $\frac { 7 } { 6 }$ 
D) Converges to $\frac { 7 } { 2 }$ 
A) Converges to $\frac { 21 } { 8 }$ 
B) Diverges. 
C) Converges to $\frac { 7 } { 6 }$ 
D) Converges to $\frac { 7 } { 2 }$

Express the given decimal as a fraction. $1.441441441 \ldots$

Use summation notation to write the given series in compact form. $\frac { 1 } { 4 } - \frac { 2 } { 16 } + \frac { 3 } { 64 } - \frac { 4 } { 256 } + \cdots$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k - 9 } { k ^ { 2 } + 6 }$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 4 } { 6 ^ { k } + k }$

Use a Taylor polynomial of specified degree n to approximate the indicated quantity.Round to four decimal places. $\sqrt { 361 } ; n = 3$

Find the fifth partial sum S₅ of the given series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }$

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 3 } } { 2 ^ { k } }$

If $\sum _ { k = 1 } ^ { \infty } a _ { k } = - 4 \text { and } \sum _ { k = 1 } ^ { \infty } b _ { k } = 9 \text {, find } \sum _ { k = 1 } ^ { \infty } \left( 9 a _ { k } - 2 b _ { k } \right)$

Find a power series for the given function. $f ( x ) = \frac { x } { 2 - x }$

Suppose that nationwide,approximately 91% of all income is spent and 9% is saved.What is the total amount of spending generated by a 55 billion dollar tax rebate if savings habits do not change?

Determine the interval of absolute convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } k ^ { 2 } \left( \frac { x } { 6 } \right) ^ { k }$

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k ! x ^ { k } } { 6 ^ { k } }$

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = e ^ { 2 x } + e ^ { - 2 x }$ ; a = 0

the given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 9 ^ { k } }$

Use a Taylor polynomial of specified degree n together with term-by-term integration to estimate the indicated definite integral.Round to six decimal places $\int _ { 0 } ^ { 0.4 } e ^ { - x ^ { 2 } } d x , n = 6$

Find the Taylor series about $x = 0$ for the indefinite integral $\int \frac { x } { 1 - 5 x ^ { 2 } } d x$

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 3 } { 7 ^ { n } }$

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Exam 9: Infinite Series and Taylor Series Approximations

Express the given decimal as a fraction. 1.441441441…1.441441441 \ldots1.441441441…

Use summation notation to write the given series in compact form. 14−216+364−4256+⋯\frac { 1 } { 4 } - \frac { 2 } { 16 } + \frac { 3 } { 64 } - \frac { 4 } { 256 } + \cdots41​−162​+643​−2564​+⋯

The given series converges. ∑k=1∞k−9k2+6\sum _ { k = 1 } ^ { \infty } \frac { k - 9 } { k ^ { 2 } + 6 }∑k=1∞​k2+6k−9​

The given series converges. ∑k=1∞46k+k\sum _ { k = 1 } ^ { \infty } \frac { 4 } { 6 ^ { k } + k }∑k=1∞​6k+k4​

Use a Taylor polynomial of specified degree n to approximate the indicated quantity.Round to four decimal places. 361;n=3\sqrt { 361 } ; n = 3361​;n=3

Find the fifth partial sum S5 of the given series. ∑n=1∞(−1)n3n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }∑n=1∞​3n(−1)n​

Determine whether the given geometric series converges,and if so,find its sum. ∑n=1∞(−59)n\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }∑n=1∞​(−95​)n

Determine the radius of convergence for the given power series. ∑k=0∞kxk4k+1\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }∑k=0∞​4k+1kxk​

The given series converges. ∑k=1∞k32k\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 3 } } { 2 ^ { k } }∑k=1∞​2kk3​

Find a power series for the given function. f(x)=x2−xf ( x ) = \frac { x } { 2 - x }f(x)=2−xx​

Suppose that nationwide,approximately 91% of all income is spent and 9% is saved.What is the total amount of spending generated by a 55 billion dollar tax rebate if savings habits do not change?

Determine the interval of absolute convergence for the given power series. ∑k=0∞k2(x6)k\sum _ { k = 0 } ^ { \infty } k ^ { 2 } \left( \frac { x } { 6 } \right) ^ { k }∑k=0∞​k2(6x​)k

Determine the radius of convergence for the given power series. ∑k=0∞k!xk6k\sum _ { k = 0 } ^ { \infty } \frac { k ! x ^ { k } } { 6 ^ { k } }∑k=0∞​6kk!xk​

Find the Taylor series for the given function at the indicated point x=a. x = a \text {. }x=a. f(x)=e2x+e−2xf ( x ) = e ^ { 2 x } + e ^ { - 2 x }f(x)=e2x+e−2x ; a = 0

the given series converges. ∑k=1∞ln⁡k9k\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 9 ^ { k } }∑k=1∞​9klnk​

Use a Taylor polynomial of specified degree n together with term-by-term integration to estimate the indicated definite integral.Round to six decimal places ∫00.4e−x2dx,n=6\int _ { 0 } ^ { 0.4 } e ^ { - x ^ { 2 } } d x , n = 6∫00.4​e−x2dx,n=6

Find the Taylor series about x=0x = 0x=0 for the indefinite integral ∫x1−5x2dx\int \frac { x } { 1 - 5 x ^ { 2 } } d x∫1−5x2x​dx

Determine whether the given geometric series converges,and if so,find its sum. ∑n=0∞37n\sum _ { n = 0 } ^ { \infty } \frac { 3 } { 7 ^ { n } }∑n=0∞​7n3​

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Express the given decimal as a fraction. $1.441441441 \ldots$

Use summation notation to write the given series in compact form. $\frac { 1 } { 4 } - \frac { 2 } { 16 } + \frac { 3 } { 64 } - \frac { 4 } { 256 } + \cdots$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k - 9 } { k ^ { 2 } + 6 }$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { 4 } { 6 ^ { k } + k }$

Use a Taylor polynomial of specified degree n to approximate the indicated quantity.Round to four decimal places. $\sqrt { 361 } ; n = 3$

Find the fifth partial sum S₅ of the given series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 5 } { 9 } \right) ^ { n }$

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$

The given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { k ^ { 3 } } { 2 ^ { k } }$

Find a power series for the given function. $f ( x ) = \frac { x } { 2 - x }$

Determine the interval of absolute convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } k ^ { 2 } \left( \frac { x } { 6 } \right) ^ { k }$

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k ! x ^ { k } } { 6 ^ { k } }$

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = e ^ { 2 x } + e ^ { - 2 x }$ ; a = 0

the given series converges. $\sum _ { k = 1 } ^ { \infty } \frac { \ln k } { 9 ^ { k } }$

Use a Taylor polynomial of specified degree n together with term-by-term integration to estimate the indicated definite integral.Round to six decimal places $\int _ { 0 } ^ { 0.4 } e ^ { - x ^ { 2 } } d x , n = 6$

Find the Taylor series about $x = 0$ for the indefinite integral $\int \frac { x } { 1 - 5 x ^ { 2 } } d x$

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 3 } { 7 ^ { n } }$