Exam 8: Infinite Sequences and Series

Use series to compute $\int _ { 0 } ^ { 1 } \cos \sqrt { x } d x$ correct to four decimal places.

Determine the limit of the sequence $a _ { n } = \frac { 5 \cos n + n } { n ^ { 2 } }$ .

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { n ! x ^ { n } } { n ^ { n } }$ .

Determine whether $a _ { n } = \frac { \sqrt { n + 1 } } { 5 n + 3 }$ is increasing, decreasing, or not monotonic.

Construct an example of power series that has $[ 3,5 ]$ as its interval of convergence.

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. Find the total distance traveled by the ball.

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { n ^ { 2 } + 1 }$ .(a) Show that the series is convergent, but not absolutely convergent.(b) Calculate the sum of the first 8 terms to approximate the sum of the series.(c) Is the approximation in part (b) an overestimate or an underestimate? (d) Estimate the error involved in the approximation from part (b).

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } n ^ { - n }$ 2) $\sum _ { n = 1 } ^ { \infty } e ^ { 100 - n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } } { n ! }$

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 1 , - \frac { 1 } { 2 } , \frac { 1 } { 4 } , - \frac { 1 } { 8 } , \ldots \right\}$ .

Find the Taylor series for $x \cos x$ about the origin.

Find a power series representation for the function $f ( x ) = \frac { 1 - x } { 1 + x }$ and give its interval of convergence.

Test the following series for convergence or divergence: $5 - \frac { 5 } { 2 } + \frac { 5 } { 5 } - \frac { 5 } { 8 } + \frac { 5 } { 11 } - \frac { 5 } { 14 } + \cdots$ .

Find the second-degree Taylor polynomial of the function $f ( x ) = x e ^ { x } \text { at } a = - 1$ .

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { n - 1 } { 5 n + 1 }$ is convergent or divergent. If it is convergent, find the sum.

Use the 3rd-degree Taylor polynomial of $f ( x ) = \sqrt { x }$ about $x = 4$ to approximate $\sqrt { 6 }$ . Use the remainder term to give an upper bound for the error in this approximation.

A sequence is defined by $b _ { n } = 1.0001 ^ { n }$ .(a) Calculate $b _ { 10 ^ { 3 } }$ and $b _ { 10 ^ { 5 } }$ .(b) Determine whether $b _ { n }$ converges or diverges. If it converges, find the limit.

Given $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } }$ .(a) Approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 5 } }$ 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?

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

Determine whether the given series is convergent or divergent. Indicate the test you use and show any necessary computation.(a) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } + 1 } { 5 n ^ { 3 } - n + 4 }$ (e) $\sum _ { n = 1 } ^ { \infty } \tan ^ { - 1 } n$ (i) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 2 ^ { n } + 3 ^ { n } }$ (b) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 + \sin n } { n } \right) ^ { 2 }$ (f) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \sqrt { 1 + \ln n } }$ (j) $\sum _ { n = 1 } ^ { \infty } \ln \left( 1 + \frac { 1 } { n } \right)$ (c) $\sum _ { n = 1 } ^ { \infty } n \cdot \sin \left( \frac { 1 } { n } \right)$ (g) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { ( n + 1 ) ^ { 3 } }$ (k) $\sum _ { n = 1 } ^ { \infty } n \cdot e ^ { - n ^ { 2 } }$ (d) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { n \sqrt { n } } + \frac { 3 } { n ^ { 3 } } \right)$ (h) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { n }$

Find the coefficient of $x ^ { 2 }$ in the Maclaurin series for $f ( x ) = 1 / ( x + 2 )$ .