Exam 8: Infinite Sequences and Series

Use the binomial series formula to obtain the Maclaurin series for $f ( x ) = ( 1 + x ) ^ { 1 / 3 }$ .

Express $\int _ { 0 } ^ { x } \frac { t } { 1 - t ^ { 5 } } d t$ as a Maclaurin Series.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 3 ^ { 2 n } } { 2 ^ { 3 n + 1 } }$ is convergent or divergent. If it is convergent, find its sum.

Let $f ( x ) = \frac { x } { 1 + \frac { 1 } { 4 } x ^ { 2 } }$ , compute $f ^ { ( 23 ) } ( 0 )$ .

Which of the following is the power series centered at $a = 0$ for $f ( x ) = \frac { 1 } { 1 - 4 x ^ { 2 } }$ ? 1) $\sum _ { n = 0 } ^ { \infty } 4 ^ { n } x ^ { 2 n }$ 2) $\sum _ { n = 0 } ^ { \infty } ( 2 x ) ^ { 2 n }$ 3) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n }$

Which one of the following series diverges?

Let $\sum a _ { n }$ and $\sum b _ { n }$ be two series. Determine whether each of the following statements is true or false. Justify your answer. (a) If $\sum a _ { n }$ converges, then $a _ { n } \rightarrow 0$ . (b) If $a _ { n } \rightarrow 0$ , then $\sum a _ { n }$ converges. (c) If $\sum a _ { n }$ converges, and $\sum b _ { n }$ diverges, then $\sum \left( a _ { n } + b _ { n } \right)$ diverges. (d) If $\sum a _ { n }$ diverges, and $\sum b _ { n }$ diverges, then $\sum \left( a _ { n } + b _ { n } \right)$ diverges. (e) If $\sum a _ { n }$ converges, and $\lim _ { n \rightarrow \infty } b _ { n } = 0$ then $\sum \left( a _ { n } + b _ { n } \right)$ converges.

Find the third Taylor polynomial associated with $f ( x ) = 2 + x ^ { 2 } - 5 x ^ { 3 } \text { at } a = 0$ . What is the remainder?

Determine whether each of the following series converges. Justify your answer by specifying which test you are using and showing any necessary computation. (a) $\sum _ { n = 1 } ^ { \infty } \frac { 2 + ( - 1 ) ^ { n } } { 2 ^ { n } }$ (b) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } + 1 } { 3 n ^ { 3 } - n + 2 }$ (c) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 + \sin n } { n } \right) ^ { 2 }$ (d) $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 1 \cdot 3 \cdot 5 \cdots \cdots ( 2 n - 1 ) }$ (e) $\sum _ { n = 1 } ^ { \infty } \frac { n ! } { n ^ { n } }$ (f) $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln n } { n } \right) ^ { n }$ (g) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \sqrt { 1 + \ln n } }$ (h) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { ( n + 1 ) ^ { 3 } }$ (i) $\sum _ { n = 1 } ^ { \infty } 3 ^ { n } \sin \left( \frac { \pi } { 4 ^ { n } } \right)$ (j) $\sum _ { n = 1 } ^ { \infty } \tan ^ { - 1 } n$ (k) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { \ln n } { n }$ (l) $\sum _ { n = 1 } ^ { \infty } \ln \left( 1 + \frac { 1 } { n } \right)$ (m) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } n ! } { 2 \cdot 5 \cdot 8 \cdot \ldots ( 3 n - 1 ) }$ (n) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n + 1 } - \sqrt { n } } { n }$

Find the sum of the series $\sum _ { n = 0 } ^ { \infty } 3 \left[ \left( \frac { 1 } { 2 } \right) ^ { n } + \left( - \frac { 1 } { 2 } \right) ^ { n } \right]$ .

Determine the limit of the sequence $a _ { n } = \frac { \sin n } { \sqrt { n } }$ .

Let $X = \{ 1,2,3 , \ldots , n , \ldots \}$ be a discrete random variable with probability density function $f ( n ) = r ( 1 - r ) ^ { n - 1 }$ , where $0 < r < 1$ .(a) Show that $\sum _ { n = 1 } ^ { \infty } f ( n ) = 1$ . Explain the significance of the value 1.(b) The expected value of the random variable X is defined by $E ( X ) = \sum _ { n = 1 } ^ { \infty } n f ( n )$ . Show that $E ( X ) = \frac { 1 } { r }$ . The distribution of X is known as the geometric distribution.

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 4 ^ { n } }$ .(a) Show that the series is absolutely convergent.(b) How many terms of the series do we need to add in order to find the sum to within 0.002? (c) What is the approximation sum in part (b)?

The population of a certain colony is 1000 and is increasing by 2% each year.(a) If $P _ { n }$ is the population after n years, find a formula for $P _ { n }$ .(b) What does the population approach as time goes on?

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

Find $\sum _ { n = 2 } ^ { \infty } \left( \sum _ { m = 2 } ^ { \infty } \frac { 1 } { n ^ { m } } \right)$ .

Which of the following are alternating series? 1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { 2 n + 1 } } { n }$ 3) $\sum _ { n = 0 } ^ { \infty } \cos ( 3 n \pi )$

Find the terms in the Maclaurin series for the function $f ( x ) = e ^ { - x }$ , as far as the term in $x ^ { 3 }$ .

Construct an example of a power series that has $( - 3,5 )$ as its interval of convergence.

If the Maclaurin series for $f ( x )$ is $1 - 9 x + 16 x ^ { 2 } - 25 x ^ { 3 } + \cdots$ , find $f ^ { 3 } ( 0 )$ .