Exam 10: Infinite Sequences and Series

Use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. - $\sum _ { n = 1 } ^ { \infty } \frac { n } { n + 2 }$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = \frac { ( n + 7 ) ! } { ( 7 n + 1 ) ! }$

Find the smallest value of N that will make the inequality hold for all n > N. - $a _ { n } = \frac { \tan ^ { - 1 } n } { e ^ { n } }$

Determine if the geometric series converges or diverges. If it converges, find its sum. - $1 + \frac { 2 } { 5 } + \left( \frac { 2 } { 5 } \right) ^ { 2 } + \left( \frac { 2 } { 5 } \right) ^ { 3 } + \left( \frac { 2 } { 5 } \right) ^ { 4 } + \ldots$

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n + 1 } } - \frac { 1 } { \sqrt { n + 2 } } \right)$

Find a formula for the nth term of the sequence. - $0,0,2,2,0,0,2,2$ (alternating 0's and 2's in pairs)

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } \frac { 8 } { 7 ^ { n } }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 0 } ^ { \infty } e ^ { - 4 n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \sqrt [ n ] { 6 ^ { n } \cdot n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { 5 + ( - 1 ) ^ { n } } { 5 }$

Determine if the series converges or diverges; if the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n + 1 } } { 8 ^ { n - 1 } }$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { n + 1 } = \frac { n a _ { n } } { n + 5 }$

Provide an appropriate response. -It can be shown that $\lim _ { \mathrm { n } \rightarrow\infty } \frac { \ln \mathrm { n } } { \mathrm { n } ^ { \mathrm { c } } } = 0$ for $\mathrm { c } > 0$ . Find the smallest value of $\mathrm { N }$ such that $\left| \frac { \ln \mathrm { n } } { \mathrm { n } ^ { \mathrm { c } } } \right| < \varepsilon$ for all $\mathrm { n } > \mathrm { N }$ if $\varepsilon = 0.01$ and $c = 1.8$ .

Express the number as the ratio of two integers. - $1.818181 \ldots$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \left( 1 + \frac { 2 } { n } \right) ^ { n }$

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { n + 1 } = 6 a _ { n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \left( \frac { 2 n } { n + 1 } \right) ^ { n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = n - \sqrt { n ^ { 2 } - 18 n }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { n ! } { 5 ^ { n } \cdot 3 ^ { n } }$

Find the limit of the sequence if it converges; otherwise indicate divergence. - $a _ { n } = \frac { ( \ln n ) ^ { 2 } } { \sqrt { n } }$