Exam 11: Infinite Sequences and Series

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } n \sin \left( \frac { \pi } { 9 n } \right)$

Which of the following series is convergent?

Let $a _ { k } = f ( k )$ where f is a continuous, positive, and decreasing function on $[ n , \infty ) ,$ and suppose that $\sum _ { k = 1 } ^ { \infty } a _ { k }$ is convergent. Defining $R _ { n } = S - S _ { n } ,$ where $S = \sum _ { n = 1 } ^ { \infty } a _ { n }$ and $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ we have that $\int _ { n + 1 } ^ { \infty } f ( x ) d x \leq R _ { n } \leq \int _ { n } ^ { \infty } f ( x ) d x$ Find the maximum error if the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 8 } { n ^ { 2 } }$ is approximated by $S _ { 40 }$

Find the interval of convergence of the series. $\sum _ { n = 2 } ^ { \infty } \frac { 6 x ^ { n } } { 5 \sqrt { n } }$

Determine whether the series is convergent or divergent by expressing $S _ { n }$ as a telescoping sum. If it is convergent, find its sum. $\sum _ { n = 2 } ^ { \infty } \frac { 3 } { n \left( n ^ { 2 } - 1 \right) }$ .

Evaluate the indefinite integral as a power series. $\int \tan ^ { - 1 } \left( t ^ { 2 } \right) d t$

Determine whether the sequence defined by $a _ { n } = \left( 1 + \frac { 5 } { n } \right) ^ { 1 / n }$ converges or diverges. If it converges, find its limit.

Which of the given series are absolutely convergent?

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

How many terms of the series $\sum _ { m = 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within 0.02?

Test the series for convergence or divergence. $\sum _ { n = 3 } ^ { \infty } \cos n$

A rubber ball is dropped from a height of 5 m onto a flat surface. Each time the ball hits the surface, it rebounds to 50% of its previous height. Find the total distance the ball travels.

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

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is $9$ . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } n ^ { 7 } } { 8 ^ { n } }$

Find the sum of the series. $\frac { 2 } { 1.3 } - \frac { 2 ^ { 2 } } { 2.3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3.3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4.3 ^ { 4 } } + \ldots$

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } n \sin \left( \frac { \pi } { 3 n } \right)$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 n ^ { 3 } + n } { \sqrt { 7 n ^ { 7 } + \ln n } }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$

Determine whether the sequence defined by $a _ { n } = \frac { 5 ^ { n } } { 8 ^ { n } + 1 }$ converges or diverges. If it converges, find its limit.