Exam 11: Infinite Sequences and Series

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 6 } + 4 } { n ^ { 8 } + n }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 n ^ { 3 } + n } { \sqrt { 7 n ^ { 7 } + \ln n } }$

Use the Integral Test to determine whether the series is convergent or divergent. $\frac { 4 } { 1 } + \frac { 4 } { 2 } + \frac { 4 } { 5 } + \frac { 4 } { 10 } + \frac { 4 } { 17 } + \cdots$

Determine the number of terms sufficient to obtain the sum of the series accurate to three decimal places. $\sum _ { n = 0 } ^ { \infty } \frac { ( - 4 ) ^ { n + 3 } } { ( n + 1 ) ! }$

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $a _ { n } = \frac { 1 } { 5 n + 1 }$ a. increasing b. decreasing c. not monotonic d. bounded

Determine which series is convergent.

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$

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 $a _ { 1 } = 20 , a _ { 2 } = 20$ . Use your calculator to guess the limit of the sequence.

Find the Taylor series for $f ( x )$ centered at the given value of a. Assume that f has a power series expansion. Also find the associated radius of convergence. $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 1 , a = 1$

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

Find the values of p for which the series is convergent. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { \gamma } }$

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

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

Determine whether the series is convergent or divergent. $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { \ln \left( n ^ { 4 } \right) }$

For which positive integers k is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( k n ! ) }$ convergent?

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$