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Exam 9: Sequences

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Question 91

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$\text { Consider the function given by } f ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( x - 7 ) ^ { n } } { n } \text {. Find the interval of }$ convergence for $\int f ( x ) d x$ .

A) $[ - 7,7 ]$
B) $[ 6,8 ]$
C) $( 6,8 )$
D) $( 0,8 )$
E) $( - 7,7 )$

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Question 92

Multiple Choice

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Find a geometric power series for the function $\frac { 1 } { 7 - x } \text { centered at } 0 \text {. }$

A)
$\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 7 ^ { n } + 1 }$
B)
$\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 7 } \right) ^ { n } + 1$
C)
$\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 7 } \right) ^ { n }$
D)
$\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 7 ^ { n + 1 } }$
E)
$\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 7 } \right) ^ { n + 1 }$

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Question 93

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$\text {The series } \sum _ { n = 1 } ^ { \infty } \frac { 6 n } { 4 n ^ { 2 } + 1 } \text { diverges. }$

Use the Root Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 7 n ^ { 2 } + 1 } { 10 n ^ { 2 } - 1 } \right) ^ { n }$

Find the fourth degree Taylor polynomial centered at $c = 7$ for the function. $f ( x ) = \ln x$

Use the Integral Test to determine the convergence or divergence of the series. $\sum _ { n = 2 } ^ { \infty } \frac { 10 } { n \sqrt { \ln n } }$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $\sum _ { n = 0 } ^ { \infty } \left( \frac { x } { 6 } \right) ^ { n }$

Find the sum of the convergent series $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 6 ^ { n } } - \frac { 1 } { 7 ^ { n } } \right)$

Suppose you go to work at a company that pays $0.08 for the first day, $0.16 for the second day, $0.32 for the third day, and so on. If the daily wage keeps doubling, what would your Total income be for working 29 days? Round your answer to two decimal places.

$\text { Graph the sequence } a _ { n } = 4 ( - 1 ) ^ { n } \text {. }$

Determine the convergence or divergence of the series. $8 \cdot \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.15 } }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 10 } \right) ^ { n }$

Suppose the winner of a $4,000,000 sweepstakes will be paid $100,000 per year for 40 years, starting a year from now. The money earns 5% interest per year. The present value of the winnings is $\sum _ { n = 1 } ^ { 40 } 100,000 \left( \frac { 1 } { 1.05 } \right) ^ { n }$ . Compute the present value using the formula for the $n$ th partial sum of a geometric series. Round your answer to two decimal places.

Use the Direct Comparison Test (if possible) to determine whether the series $\sum _ { n = 9 } ^ { \infty } \frac { 1 } { n ^ { 5 / 6 } - 8 }$

Use the Limit Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } 2 \sin \frac { 1 } { n }$ .

Use the Direct Comparison Test to determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 7 n ^ { 2 } + 9 }$ .

$\text { Use the series } \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 1 } \text { for } f ( x ) = \arctan x \text { to approximate the value of }$ $\arctan \frac { 1 } { 8 }$ using $R _ { N } \leq 0.001$ . Round your answer to three decimal places.

Suppose the annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so on. Summing all of this spending indefinitely, leads to the geometric series $\sum _ { i = 0 } ^ { \infty } 100 \left( 0.75 ^ { i } \right)$ . Find the sum of this series.

$\text { Use a graphing utility to graph } f ( x ) = \frac { 12 } { \sqrt [ 3 ] { x } } \text { and } P _ { 1 } \text {, a first-degree polynomial }$ function whose value and slope agree with the value and slope of f at .