Exam 8: Infinite Sequences and Series

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } { n } x ^ { n }$ .

If $a _ { 1 } = 1$ and $a _ { n + 1 } = \sqrt { 1 + a _ { n } }$ for $n \geq 1$ and $\lim _ { n \rightarrow \infty } a _ { n } = L$ is assumed to exist, then what must $L$ be?

If $\frac { 3 n - 1 } { n + 1 } < x _ { n } < \frac { 3 n ^ { 2 } + 6 n + 2 } { n ^ { 2 } + 2 n + 1 }$ for all positive integers n, then find $\lim _ { n \rightarrow \infty } x _ { n }$ .

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 2 n ^ { 2 } }$ .

Find the values of x for which the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { x ^ { 2 } + 2 } \right) ^ { n }$ converges.

Which one of the following series diverge?

Determine the limit of the sequence $a _ { n } = \frac { n ! } { ( n + 3 ) ! }$ .

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 2 \cdot 5 \cdot 8 \cdot \ldots \cdot ( 3 n - 1 ) } { 6 ^ { n } n ! }$ converges or diverges by the Ratio Test or Root Test.

Find the interval of convergence of the power series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 2 ) ^ { n } } { \sqrt [ 3 ] { n } }$ .

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n } } + \frac { 2 } { n ^ { 3 } } \right)$ 2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { n } - \frac { 1 } { n + 1 } \right)$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos ( 1 / n ) } { n ^ { 2 } }$

A sequence is defined by $a _ { n } = r ^ { n }$ , where r is a constant. For what values of r will the sequence converge? What is the limit?

Given $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ .(a) Approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } }$ by using the sum of the first 4 terms.(b) Estimate the error involved in the approximation in part (a).(c) How many terms are required to ensure that the sum is accurate to within 0:001?

Consider the sequence defined by $a _ { n } = ( - 1 ) ^ { n }$ . (n starts at 1) (a) Write the first five terms of the sequence.(b) Determine the limit of the sequence.(c) Let $b _ { n } = \frac { a _ { n + 1 } } { a _ { n } }$ . Write the first five terms of this sequence.(d) Determine the limit of $b _ { n }$ .(e) Let $c _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ . Write the first five terms of this sequence.(f) Determine the limit of $c _ { n }$ .

If $a _ { 1 } = 1$ and $a _ { n + 1 } = 3 - \left( \frac { 1 } { a _ { n } } \right)$ for $n \geq 1$ , find the limit of the sequence $a _ { n }$ .

Show that the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { 4 n }$ is convergent. How many terms of the series do we need to add to find the sum to within 0.01?

Given that the power series for $\frac { 1 } { 1 - x }$ is $\sum _ { n = 0 } ^ { \infty } x ^ { n }$ , find the power series for $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ in terms of powers of x.

Find a power series representation for the function $f ( x ) = \arctan ( 3 x )$ and determine its radius of convergence.

For which of the following series will the Ratio Test fail to give a definite answer (i.e., be inconclusive)? 1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 99 } { 100 } \right) ^ { n }$ 2) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 100 } { 99 } \right) ^ { n }$ 3) $\sum _ { n = 1 } ^ { \infty } n ^ { - 100 }$

Find the interval of convergence for $\sum _ { k = 0 } ^ { \infty } \left( \frac { e ^ { k } } { k + 1 } \right) x ^ { k }$ .

Here is a two-player game: Two players take turns tossing a fair die. The first player to get a 5 wins the game. What is the probability that the player who starts first wins the game?