Exam 11: Infinite Sequences and Series

Determine the number of terms sufficient to obtain the sum of the series accurate to three decimal places. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 5 ) ^ { n + 3 } } { ( n + 1 ) ! }$

Find a formula for the general term $a _ { n }$ of the sequence, assuming that the pattern of the first few terms continues. $\left\{ - \frac { 1 } { 2 } , \frac { 16 } { 3 } , - \frac { 81 } { 4 } , \frac { 256 } { 5 } , - \frac { 625 } { 6 } , \ldots \right\}$

Determine whether the series is convergent or divergent. $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \ln \left( n ^ { 9 } \right) }$

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

A right triangle ABC is given with $\theta = 1.1$ and $| \mathrm { AC } | = b = 12$ . CD is drawn perpendicular to AB, DE is drawn perpendicular to BC, EF $\perp$ AB and this process is continued indefinitely as shown in the figure. Find the total length of all the perpendiculars $| \mathrm { CD } | + | \mathrm { DE } | + | \mathrm { EF } | + | \mathrm { FG } | + \ldots$ Write your answer to two decimal places.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { 4 ^ { m } m ^ { 3 } } { m ! }$

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 10 }$

Find the partial sum $S _ { 7 }$ of the series $\sum _ { m - 1 } ^ { \infty } \frac { 4 } { 10 + 7 ^ { m } }$ . Give your answer to five decimal places.

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { \sqrt { n } }$

Express the number $0 . \overline { 84 }$ as a ratio of integers.

Find the values of p for which the series is convergent. $\sum _ { n = 1 } ^ { \infty } 4 n \left( 1 + n ^ { 2 } \right) ^ { p }$

Approximate the sum to the indicated accuracy. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 4 ^ { n } n ! }$

Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $S _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$ .

Determine whether the given series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { e ^ { 1 / n ^ { 7 } } } { n ^ { 8 } }$

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Find the Maclaurin series for f and its radius of convergence. $f ( x ) = 2 \ln ( 1 - x )$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \sqrt [ 3 ] { n } }$

Find the exact value of the limit of the sequence defined by $a _ { 1 } = \sqrt { 4 } , a _ { n + 1 } = \sqrt { 4 + a _ { n } }$ .

Determine whether the series converges or diverges. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \ln \left( n ^ { 8 } \right) }$