Exam 9: Sequences

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.

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

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

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

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

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

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

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

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.

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 $\sum _ { n = 1 } ^ { \infty } 2 \sin \frac { 1 } { n }$ .

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.

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

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

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

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

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

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