Exam 10: Infinite Sequences and Series

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 3 ^ { 1 / ( n + 1 ) } } - \frac { 1 } { 3 ^ { 1 / n } } \right)$

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

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 2 ^ { n } } - \frac { 1 } { 4 ^ { n } } \right)$

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

A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. - $a _ { 1 } = 1 , a _ { 2 } = 4 , a _ { n + 2 } = a _ { n + 1 } + a _ { n }$

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \frac { 7 } { ( 4 n - 1 ) ( 4 n + 3 ) }$

Find a formula for the nth term of the sequence. - $0 , - 1,0,1,0 , - 1,0,1 ( 0 , - 1,0,1$ repeated)

Determine if the series converges or diverges. If the series converges, find its sum. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n + 1 } } - \frac { 1 } { \sqrt { n + 3 } } \right)$

Find the sum of the series. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { 1 } { 7 ^ { n } } - \frac { ( - 1 ) ^ { n } } { 7 ^ { n } } \right)$

Provide an appropriate response. -A sequence of rational numbers $\left\{ \mathrm { r } _ { \mathrm { n } } \right\}$ is defined $b \mathrm { y } \mathrm {} \mathrm { r } _ { 1 } = \frac { 2 } { 1 }$ , and if $\mathrm { r } _ { \mathrm { n } } = \frac { \mathrm { a } } { \mathrm { b } }$ then $\mathrm { r } _ { \mathrm { n } } + 1 = \frac { \mathrm { a } + \mathrm { b } } { \mathrm { a } - \mathrm { b } }$ . Find $\mathrm { r } _ { 50 }$ .

Find a formula for the nth term of the sequence. - $1 , - \frac { 1 } { 4 } , \frac { 1 } { 9 } , - \frac { 1 } { 16 } , \frac { 1 } { 25 }$ (reciprocals of squares with alternating signs)

Find the values of x for which the geometric series converges. - $\sum _ { n = 0 } ^ { \infty } 10 ^ { n } x ^ { n }$

Find a formula for the nth term of the sequence. - $8,9,10,11,12$ (integers beginning with 8 )

Find the sum of the geometric series for those x for which the series converges. - $\sum _ { n = 0 } ^ { \infty } \left( \frac { x - 9 } { 6 } \right) ^ { n }$

Express the number as the ratio of two integers. - $0.22222 \ldots$

Find the values of x for which the geometric series converges. - $\sum_{n=0}^{\infty}|\sin 5 x|^{n}$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $3 - 24 + 192 - 1536 + \ldots + ( - 1 ) ^ { n - 1 } 3 \cdot 8 ^ { n - 1 } + \ldots$

Determine if the sequence is monotonic and if it is bounded. - $a _ { n } = 4 - \frac { 1 } { n }$

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. - $\frac { 9 } { 1 \cdot 2 \cdot 3 } + \frac { 9 } { 2 \cdot 3 \cdot 4 } + \frac { 9 } { 3 \cdot 4 \cdot 5 } + \ldots + \frac { 9 } { n ( n + 1 ) ( n + 2 ) } + \ldots$

Express the number as the ratio of two integers. - $0.3131313 \ldots$