Exam 8: Infinite Sequences and Series

Approximate the definite integral $\int _ { 0 } ^ { 1 } e ^ { - x ^ { 2 } } d x$ accurate to six decimal places.

Determine the limit of the sequence $a _ { n } = \frac { \sqrt { n + 1 } - \sqrt { n } } { \sqrt { n + 1 } + \sqrt { n } }$ .

Find the interval of convergence for $\sum _ { n = 1 } ^ { \infty } \frac { x ^ { n } } { 4 n ^ { 2 } }$ .

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

Use the binomial series to expand $( 4 + x ) ^ { 3 / 2 }$ as a power series. State the radius of convergence.

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( n + 2 ) }$ .

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

Write the Taylor polynomial at 0 of degree 4 for $f ( x ) = \ln ( 1 + x )$ .

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

Determine whether the series $\sum _ { n = 1 } ^ { \infty } ( 0.9999 ) ^ { n }$ is convergent or divergent. If it is convergent, find the sum.

Use the binomial series to expand $\frac { 1 } { \sqrt { 2 + x } }$ as a power series. State the radius of convergence.

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

Show that the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { 3 n + 1 }$ is convergent. How many terms of the series do we need to add to find the sum to within 0.01?

Express the number 1.363636... as a ratio of integers.

Show that if $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } \cos \left( a _ { n } \right)$ diverges.

Evaluate $\lim _ { n \rightarrow \infty } \left( 1 - \frac { 2 } { n } \right) ^ { n }$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } 3 x ^ { n }$ .

Consider the sequence defined by $a _ { n } = \frac { n ! } { 2 \cdot 5 \cdot 8 \cdots ( 3 n - 1 ) } , n \geq 1$ (a) Evaluate the first three terms of this sequence.(b) Find the limit.

Which of the following series are convergent, but not absolutely convergent? 1) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt [ 3 ] { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \sqrt { n } \ln n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { \cos n } { 2 ^ { n } }$

Approximate the definite integral $\int _ { 0 } ^ { 0.1 } \frac { 1 } { x ^ { 5 } + 1 } d x$ accurate to six decimal places.