Consider the sequence $a_n=\frac{n^3{e}^{-n}}{8^n},n=1,2,$dots

$($a$)$ Show that the sequence is monotonic.

$($b$)$ Show that the sequence is bounded.

$($c$)$ Is the sequence convergent? If so$,$ find its limit$.$

Find the sum of the series

${\displaystyle \underset{n=1}{\overset{}{}}}[ta{n}^{-1}(n+1)-ta{n}^{-1}(n)+\frac{(-3{)}^{n-1}}{2^{3n}}]$

Use the integral test to determine whether the series ${\displaystyle \underset{k=1}{\overset{}{}}}\frac{1}{k[(lnk{)}^2+4]}$ converges or diverges.

Determine whether each of the following series

$($a$){\displaystyle \underset{n=2}{\overset{}{}}}(-1{)}^n(\sqrt[\phantom{2}]{n+1}-\sqrt[\phantom{2}]{n})$

$($b$){\displaystyle \underset{n=1}{\overset{}{}}}\frac{ta{n}^{-1}(n)}{3n^2+7}$

$($c$){\displaystyle \underset{k=1}{\overset{}{}}}\frac{k^{\frac{4}{3}}+5}{8k^2+\sqrt[3]{k}+1}$ is absolutely convergent, conditionally convergent, or divergent.

Determine whether the series

$($a$){\displaystyle \underset{n=1}{\overset{}{}}}\frac{(n+1)!}{e^{n^2}}$

$($b$){\displaystyle \underset{n=1}{\overset{}{}}}(\frac{-2n}{n+1}{)}^{5n}.$

are convergent or divergent.

Find the radius of convergence and the interval of convergence of the power series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{2^n}{5^{n+1}(n+1)}(x-3{)}^n$

Find the Taylor series of $f(x)=\frac{1}{x^2},$ centered at $a=1.$