Exam 8: Infinite Sequences and Series

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D). 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 2 }$

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n } }$ .

Estimate the range of values of x for which the approximation $\sqrt { x ^ { 2 } + 3 } = 2 + \frac { 1 } { 2 } ( x - 1 ) + \frac { 3 } { 16 } ( x - 1 ) ^ { 2 }$ is accurate to within 0.0002.

Find the coefficient of $x ^ { 4 }$ in the Maclaurin series for $f ( x ) = e ^ { - 2 x }$ .

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

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { ( n + 9 ) ( n + 10 ) } { n ( n + 1 ) }$ .

The first three derivatives of $f ( x ) = ( x + 4 ) ^ { 3 / 2 }$ are $f ^ { \prime } ( x ) = \frac { 3 ( x + 4 ) ^ { 1 / 2 } } { 2 }$ , $f ^ { \prime \prime } ( x ) = \frac { 3 } { 4 ( x + 4 ) ^ { 1 / 2 } }$ and $f ^ { \prime \prime } ( x ) = \frac { - 3 } { 8 ( x + 4 ) ^ { 3 / 2 } }$ .(a) Give the first four terms of the Taylor series associated with f at $a = - 3$ .(b) Give the second-order Taylor polynomial, $T _ { 2 } ( x )$ , associated with f at $a = 0$ .(c) Suppose that $x \geq 0$ and that $T _ { 2 } ( x )$ from part (b) is used to approximate $f ( x )$ . Prove that the error in this approximation does not exceed $\frac { x ^ { 3 } } { 128 }$ .

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

Find the values of x for which the series $\sum _ { n = 1 } ^ { \infty } ( x - 1 ) ^ { n }$ converges.

Which of the following is the degree 2 Taylor polynomial centered at $a = 3$ for $f ( x ) = \ln x$ ? 1) $\ln 3 + \frac { x - 3 } { 3 } - \frac { ( x - 3 ) ^ { 2 } } { 18 }$ 2) $\ln 3 + \frac { x - 3 } { 3 } - \frac { ( x - 3 ) ^ { 2 } } { 9 }$ 3) $\ln 3 - \frac { x - 3 } { 3 } + \frac { ( x - 3 ) ^ { 2 } } { 18 }$

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

If a sequence is bounded, does the sequence necessarily have a limit? Explain.

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

Find the radius of convergence of the Maclaurin series for $f ( x ) = \frac { 1 } { 4 + x ^ { 2 } }$ .

Determine whether $a _ { n } = \sin \left( \frac { n \pi } { 2 } \right)$ converges or diverges. If it converges, find the limit.

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \left( \frac { n } { 2 + 3 n } \right) ^ { n }$ 2) $\sum _ { n = 2 } ^ { \infty } \frac { n + 1 } { \sqrt { n ^ { 4 } - 1 } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + n ^ { 2 } }$

A rubber ball is dropped from a height of 10 feet and bounces to $\frac { 3 } { 4 }$ its height after each fall. If it continues to bounce until it comes to rest, find the total distance in feet it travels.

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 3 ^ { n } + 2 ^ { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { 3 ^ { n } } { n + 5 ^ { n } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 1 + 4 n }$

Tell which of the following three series cannot be found convergent by the Ratio Test but can be found convergent by comparison with a p-series. 1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n } } { n ^ { 2 } + n }$

The series $\sum _ { n = 0 } ^ { \infty } r ^ { n }$ converges if and only if