Exam 11: Infinite Sequences and Series

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

Use the sum of the first 9 terms to approximate the sum of the following series. $\sum _ { n = 2 } ^ { \infty } \frac { 6 } { n ^ { 5 } + n ^ { 4 } }$ Write your answer to six decimal places.

Find the sum of the series. $\sum _ { n = 2 } ^ { \infty } 2 n ( n - 1 ) x ^ { n + 1 } \quad | x | < 1$

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

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

Find a power series representation for the indefinite integral. $\int \frac { \sin 9 x } { x } d x$

When money is spent on goods and services, those that receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending D dollars. Suppose that each recipient of spent money spends $100 c \%$ and saves $100 s \%$ of the money that he or she receives. The values c and s are called the marginal propensity to consume and the marginal propensity to save and, of course, $c + s = 1$ . The number k = 1/s is called the multiplier. What is the multiplier if the marginal propensity to consume is $60 \%$ ?

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 n ^ { 2 } + 5 } { 2 n ^ { 2 } + 6 }$

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

Graph the sequence defined by $a _ { n } = ( - 1 ) ^ { n } \frac { 2 n + 1 } { n + 2 }$ . Use the graph to guess at the convergence or divergence of the sequence.

Determine whether the sequence defined by $a _ { n } = \left( 1 + \frac { 8 } { n } \right) ^ { 1 / n }$ converges or diverges. If it converges, find its limit.

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

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

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 3 ^ { n } }$

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

Test the series for convergence or divergence. $\sum _ { k = 1 } ^ { \infty } \frac { ( - 6 ) ^ { k + 1 } } { 7 ^ { 2 k } }$

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

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

Determine whether the series is convergent or divergent by expressing $S _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 14 } { n ^ { 2 } + 2 n }$ .

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ! n }$