Exam 8: Infinite Sequences and Series

Consider the series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { ( 4 n ) ! }$ .(a) Show that $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { ( 4 n ) ! }$ is absolutely convergent.(b) Calculate the sum of the first 3 terms to approximate the sum of the series.(c) Estimate the error involved in the approximation from part (b).

Determine whether $\sum _ { n = 1 } ^ { \infty } 3 n e ^ { - n ^ { 2 } }$ converges or diverges.

Find the terms in the power series expansion for the function $f ( x ) = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$ , as far as the term in $x ^ { 3 }$ .

Find a formula for the general term $a _ { n }$ of the sequence $\{ 1,6,120,5040 , \ldots \}$ .

If $\sum _ { n = 1 } ^ { \infty } c _ { n } ( x - n ) ^ { n }$ is convergent at $x = 5$ , what can be said about the convergence or divergence of the following series? (a) $\sum _ { n = 1 } ^ { \infty } c _ { n }$ (b) $\sum _ { n = 1 } ^ { \infty } c _ { n } ( - 2 ) ^ { n }$ (c) $\sum _ { n = 1 } ^ { \infty } c _ { n } 4 ^ { n }$

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 1 } { n ^ { 4 } }$ .(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 0.001? (c) What is the approximation sum in part (b)?

Determine the limit of the sequence $a _ { n } = \frac { ( - 2 ) ^ { n } } { n }$ .

Find the terms of the Maclaurin series for $\frac { 1 } { \sqrt { 1 - x } }$ , as far as the term in $x ^ { 3 }$ .

Consider the function $f ( x ) = 3 x ^ { 4 } - 24 x ^ { 3 } + 72 x ^ { 2 } - 96 x + 49$ .(a) Find the fourth-degree Taylor polynomial of f at $a = 2$ .(b) What is the remainder? (c) What is the absolute minimum value of f, and where does it occur?

A superball is dropped from a height of 8 ft. Each time it strikes the ground after falling from a height of t ft. it rebounds to a height of $\frac { 3 } { 4 } t$ feet. How long does it take for the ball to come to rest? (Use $g = 32 \mathrm { ft } / \mathrm { s } ^ { 2 }$ .)

Find the coefficient of $x ^ { 4 }$ in the Maclaurin series for $f ( x ) = x \cos \left( x ^ { 3 } \right)$ .

Which of the following three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 2 ) }$ converges? 1) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 2 }$ 2) Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 2 }$ 3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } 3 n ^ { - 1 }$

Determine whether $\sum _ { n = 1 } ^ { \infty } \frac { \cos n + 3 ^ { n } } { n ^ { 2 } + 5 ^ { n } }$ is convergent or divergent.

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { \ln n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n }$

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n - 1 } } { 4 ^ { n } }$ .

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

Find the limit of the sequence $a _ { n } = \frac { \ln ( 3 n ) } { \ln n }$ .

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ! }$ .

Use the binomial series to expand the function $\sqrt { 4 + x }$ as a power series. Give the coefficient of $x ^ { 2 }$ in that series.

Consider the two series: (a) $\sum _ { k = 2 } ^ { \infty } \frac { \ln k } { k }$ and (b) $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k \ln k }$ . Suppose you compare (a) and (b) to the series $\sum _ { k = 1 } ^ { \infty } \frac { 1 } { k }$ . What (if anything) can you conclude about the convergence or divergence of (a) and (b) using only the Comparison Test?