Exam 11: Infinite Sequences and Series

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$ . Use your calculator to guess the limit of the sequence.

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

Use the sum of the first 10 terms to approximate the sum of the series.Estimate the error. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 1 + 4 ^ { n } }$

Determine whether the given series converges or diverges.If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } + 8 ^ { n } } { 12 ^ { n } }$

Determine whether the sequence defined by $a _ { n } = 5 + 8 ( - 1 ) ^ { n }$ converges or diverges. If it converges, find its limit.

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

Determine whether the geometric series converges or diverges.If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }$

How many terms of the series $\sum _ { m = 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within $0.02 ?$

Determine whether the geometric series converges or diverges.If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent?

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Find all values of $p$ for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { p } }$ converges.

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$ . Use your calculator to guess the limit of the sequence.

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. $\cos x \approx 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 24 } \quad \mid$ error $\mid < 0.08$ Write $a$ such that $- a < x < a$ .

Find a formula for the general term 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 geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$