Exam 8: Infinite Sequences and Series

Find the terms in the Maclaurin series for the function $f ( x ) = \ln ( 1 + x )$ , as far as the term in $x ^ { 3 }$ .

Use the Integral Test to determine if the following series converges or diverges: $\sum _ { n = 1 } ^ { \infty } \frac { n } { \left( n ^ { 2 } + 1 \right) ^ { 2 } }$ .

Find the radius of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { n ! } { 4 ^ { n } } ( x + 3 ) ^ { n }$ .

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n }$ 2) $\sum _ { n = 1 } ^ { \infty } 2 ^ { n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + n ^ { 3 } }$

Which of the following is the degree 2 Taylor polynomial centered at $a = - 1$ for $f ( x ) = \frac { 1 } { x }$ ? 1) $- 1 - x - x ^ { 2 }$ 2) $- 1 - ( x + 1 ) - ( x + 1 ) ^ { 2 }$ 3) $- 1 - ( x + 1 ) - 2 ( x + 1 ) ^ { 2 }$

Find the coefficient of $x ^ { 3 }$ in the binomial series for $\frac { 1 } { ( 1 + x ) ^ { 4 } }$ .

Which one of the following series diverges?

Determine whether or not $\sum _ { k = 2 } ^ { \infty } \frac { 1 } { k ( \ln k ) ^ { 2 } }$ converges.

Show that $f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { n } } { n ! }$ is a solution to the differential equation $\frac { d y } { d x } = 8 y$ .

Which of the three series below converges? 1) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n \ln n }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 2 } }$ 3) $\sum _ { n = 2 } ^ { \infty } \frac { 1 } { n ( \ln n ) ^ { 3 } }$

Use the Ratio Test to examine the two series below, stating: absolute convergence (A), divergence (D), or Ratio Test inconclusive (I). 1) $\sum _ { n = 1 } ^ { \infty } n ^ { - 100 }$ 2) $\sum _ { n = 1 } ^ { \infty } 100 ^ { - n }$

Find the sum of the series $\sum _ { n = 2 } ^ { \infty } ( n - 1 ) x ^ { n }$ , where $| x | < 1$ .

Approximate the definite integral $\int _ { 0 } ^ { 0 S } \frac { \ln ( 1 + x ) } { x } d x$ accurate to six decimal places.

Find the second-degree Taylor polynomial for $f ( x ) = \sqrt { x }$ , centered about $a = 100$ . Also obtain a bound for the error in using this polynomial to approximate $\sqrt { 100.1 }$ .

Examine the two series below for absolute convergence (A), convergence that is not absolute (C), or divergence (D). 1) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { \ln ( n + 1 ) }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { ( \ln ( n + 1 ) ) ^ { 2 } }$

Determine the limit of the sequence $a _ { n } = \frac { 5 \cos n } { n }$ .

Which of the following is a power series? 1) $1 + 2 x + 3 x ^ { 4 }$ 2) $\sum _ { n = 1 } ^ { \infty } ( 5 x + 1 ) ^ { 2 n }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { x ^ { n } }$

Find the third-degree Taylor polynomial centered at $x = 1$ for $f ( x ) = \ln x$ . Use this result to approximate $\ln 1.2$ .

According to the estimates found in the justification for the Integral Test, the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 1.001 } }$ must lie between what two values?

Which of the following series converges? 1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \sqrt { e ^ { n } } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { \sqrt [ 3 ] { e ^ { n } } }$