Exam 8: Infinite Sequences and Series

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \frac { 1 - \cos x } { x }$ ? 1) $\frac { x } { 2 } - \frac { x ^ { 3 } } { 24 }$ 2) $- \frac { x } { 2 } + \frac { x ^ { 3 } } { 24 }$ 3) $x - x ^ { 3 }$

Which one of the following series diverges?

Find the coefficient of $x$ in the binomial series for $\sqrt { 1 + x }$ .

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

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 + ( - 1 ) ^ { n } } { 3 ^ { n } }$ is convergent or divergent. If it is convergent, find the sum.

Use the binomial series to expand $\frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 3 } } }$ as a power series. State the radius of convergence.

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ - \frac { 1 } { 2 } , 0 , \frac { 1 } { 10 } , \frac { 2 } { 17 } , \frac { 3 } { 26 } , \ldots \right\}$ .

Determine whether $a _ { n } = \frac { n \cos n } { n ^ { 2 } + 1 }$ converges or diverges. If it converges, find the limit.

Find the sum of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } \pi ^ { 2 n + 1 } } { 3 ^ { 2 n } ( 2 n + 1 ) ! }$ .

Determine whether each of the following series is convergent or divergent.(a) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ (b) $\sum _ { n = 0 } ^ { \infty } n \sin \left( \frac { 1 } { n } \right)$ (c) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { 3 ^ { n } } { 5 ^ { n } }$

According to Taylor's Formula, what is the maximum error possible in the use of the sum $\sum _ { n = 0 } ^ { 4 } \frac { x ^ { n } } { n ! }$ to approximate $e ^ { x }$ in the interval $- 1 \leq x \leq 1$ ?

Find the limit of the sequence $\{ \sqrt { 3 } , \sqrt { 3 \sqrt { 3 } } , \sqrt { 3 \sqrt { 3 \sqrt { 3 } } } , \ldots \}$ .

Express the number $0.2 \overline { 15 }$ as a ratio of integers.

Determine whether the series $\sum _ { n = 1 } ^ { \infty } \cos \left( \frac { \pi } { 2 n ^ { 2 } - 1 } \right)$ is convergent or divergent. If it is convergent, find the sum.

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

Estimate the range of values of x for which the approximation $\frac { 1 } { x } = 1 - ( x - 1 ) + ( x - 1 ) ^ { 2 }$ is accurate to within 0.01.

(a) Find the third-order Taylor polynomial associated with $f ( x ) = \sin ^ { - 1 } x \text { at } a = 0$ .(b) Use the Taylor polynomial from part (a) to find an approximation of $\sin ^ { - 1 } 0.2$ .(c) Compare the value you calculated in part (b) with your calculator's value for $\sin ^ { - 1 } 0.2$

Which one of the following series diverges?

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { n 3 ^ { n } } { 4 ^ { n } }$ .

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { 1 } { 2 n - 1 }$ .(a) Show that the series is convergent, but not absolutely convergent.(b) Calculate the sum of the first 9 terms to approximate the sum of the series.(c) Is the approximation in part (b) an overestimate or an underestimate? (d) Estimate the error involved in the approximation from part (b).