Exam 8: Infinite Sequences and Series

Consider the sequence defined by $a _ { n } = \left( \frac { 3 } { 2 } \right) ^ { n }$ . (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 $b _ { n }$ .

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

Find the sum of the series $\sum _ { n = 4 } ^ { \infty } \ln \left( \frac { n + 1 } { n } \right)$ .

What is the value of p that marks the boundary between convergence and divergence of the series $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { y } }$ ?

Which of the following three tests will establish that the series $\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { 7 n ^ { 3 } + 46 } }$ diverges? 1) Limit Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ 2) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ 3) Comparison Test with $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 / 2 }$

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

Express the number $0 . \overline { 307 }$ as a ratio of integers.

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

Which of the three series below converges? 1) $\sum _ { n = 0 } ^ { \infty } \left( - \frac { 3 } { 4 } \right) ^ { n }$ 2) $\sum _ { n = 0 } ^ { \infty } \left( \frac { e } { \pi } \right) ^ { n }$ 3) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n }$

Express $f ( x ) = \int _ { 0 } ^ { x } \frac { \sin t } { t } d t$ as a Maclaurin Series.

Let $X = \{ 0,1,2,3 , \ldots , n , \ldots \}$ be a discrete random variable with probability density function $f ( n ) = e ^ { - \mu } \frac { \mu ^ { n } } { n ! }$ , where $0 < \mu$ .(a) Show that $\sum _ { n = 0 } ^ { \infty } f ( n ) = 1$ . Explain the significance of the value 1.(b) The expected value of the random variable X is defined by $E ( X ) = \sum _ { n = 0 } ^ { \infty } n f ( n )$ . Show that $E ( X ) = \mu$ . The distribution of X is known as the Poisson distribution.

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \ln ( n + 1 ) }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \ln ( n + 1 )$ 3) $1 - \frac { 1 } { 2 } + \frac { 2 } { 3 } - \frac { 3 } { 4 } + \frac { 4 } { 5 } - \frac { 5 } { 6 } + \cdots$

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

Find the Taylor series for $y = \ln x$ at 2.

Consider the sequence defined by $a _ { n } = \left( \frac { 2 } { 3 } \right) ^ { n }$ . (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 $b _ { n }$ .

Find the coefficient of $x ^ { 3 }$ in the Maclaurin series for $f ( x ) = \sin 2 x$ .

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

Tell which of the following series can be compared with geometric series to establish convergence.1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n } }$

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

Test the following series for convergence or divergence: $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n \ln n }$ .