command

eption

$2.$ Answer the same questions as above, or explain why you can't answer them, with this power series representation for a new $f(x)$ :

$5{\displaystyle \underset{k=2}{\overset{}{}}}\frac{(x+3{)}^{2k}}{9^{k+1}}$

Also, when the series above converges, what is its sum? $($This might remind you of another way of knowing when this series converges.$)$

$3.$ Determine whether each of the following series converges or diverges. Completely justify your answer.

$($a$){\displaystyle \underset{k=5}{\overset{}{}}}\frac{k-sink}{k^3+k+k!}$

$($b$){\displaystyle \underset{k=1}{\overset{}{}}}\frac{2^k+k^2}{3^k+k^3}$

$($c$){\displaystyle \underset{k=1}{\overset{}{}}}\frac{cosk}{k^2}$

$($d$){\displaystyle \underset{k=1}{\overset{}{}}}(\frac{-\frac{k}{k+1}}{2}{)}^k$

$($e$){\displaystyle \underset{k=1}{\overset{}{}}}(\frac{-\frac{k+1}{k}}{3}{)}^k$

$($f$){\displaystyle \underset{k=1}{\overset{}{}}}\frac{ln{e}^k}{2k}$

$2$