Deck 9: Infinite Series and Taylor Series Approximations

Question 1

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

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

Answer 1

$- 6 < x < 6$

Question 2

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

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) $\sum _ { n = 0 } ^ { \infty } \frac { 2 \left( 2 ^ { n } \right) } { n ! } x ^ { n }$

Answer 2

$\sum _ { n = 0 } ^ { \infty } \frac { 2 \left( 2 ^ { 2 n } \right) } { ( 2 n ) ! } x ^ { 2 n }$

Question 3

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?

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

Answer 3

$611 billion

Question 4

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

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

Question 5

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 $\int _ { 0 } ^ { 0.4 } e ^ { - x ^ { 2 } } d x , n = 6$

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

Question 6

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

Question 7

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

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

Question 8

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

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

Question 9

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

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

Question 10

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

Question 11

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

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 12

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

Question 13

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

Question 14

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

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 15

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

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

Question 16

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

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

Question 17

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

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

Question 18

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

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

Question 19

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?

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

Question 20

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

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