Exam 8: Infinite Sequences and Series

Find a power series representation for $\int \frac { x } { x ^ { 3 } + 1 } d x$ .

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

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

Determine whether the series $- \frac { 81 } { 100 } + \frac { 9 } { 10 } - 1 + \frac { 10 } { 9 } - \ldots$ is convergent or divergent. If it is convergent, find its sum.

Find the sum of the series $\sum _ { n = 3 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }$ .

Consider the sequence defined by $a _ { n } = \left( - \frac { 3 } { 4 } \right) ^ { n }$ . (n starts at 1) (a) Write the first five terms of the sequence.(b) Determine the limit of the sequence.(c) Let $b _ { n } = \frac { a _ { n + 1 } } { a _ { n } }$ . Write the first five terms of this sequence.(d) Determine the limit of $b _ { n }$ .

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \cos \left( \frac { \pi } { n } \right)$ .

A series $\sum _ { k = 1 } ^ { \infty } a _ { k }$ has partial sums, $S _ { n }$ , given by $S _ { n } = \frac { 7 n - 2 } { n }$ (a) Is $\sum _ { k = 1 } ^ { \infty } a _ { k }$ convergent? If it is, find the sum.(b) Find $\lim _ { n \rightarrow \infty } a _ { k }$ .(c) Find $\sum _ { k = 1 } ^ { 200 } a _ { k }$

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 2 n + 1 } } { ( 2 n + 1 ) ! }$ .

Find the coefficient of $x ^ { 5 }$ in the Maclaurin series for $f ( x ) = \int \cos \left( x ^ { 2 } \right) d x$ .Note: The series is unique except for the constant of integration.

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

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

Use the Ratio Test to examine the two series below, stating: absolute convergence (A), divergence (D), or Ratio Test inconclusive (I). 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 2 ^ { 2 n + 1 } } { 5 ^ { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 5 ^ { n } } { 2 ^ { 2 n + 1 } }$

Find the limit of the sequence $a _ { n } = n e ^ { 1 / n }$ .

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

Determine if the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { n ^ { 100 } }$ converges or diverges by the Ratio Test or Root Test.

Use the integral test to show that the series $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \ln k ) ^ { y } }$ converges if $p > 1$ and diverges if $p \leq 1$ .Hint: Consider the two cases $p = 1$ and $p \neq 1$ .

Determine the limit of the sequence $a _ { n } = ( - 1 ) ^ { n } \left( 1 - \frac { 1 } { \sqrt { n } } \right)$ .

Estimate the range of values of x for which the approximation $\ln x = \ln 2 + \frac { 1 } { 2 } ( x - 2 ) - \frac { 1 } { 8 } ( x - 2 ) ^ { 2 }$ is accurate to within 0.01.

Which of the following series are convergent, but not absolutely convergent? 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n + 2 } { n ^ { 2 } + 1 }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { n + 1 }$