$Q1.$

$(a)$ Determine the interval $of$ convergence $of$ the power series $h(x)={\displaystyle \underset{n=2}{\overset{}{}}}\frac{7}{3^{n+1}nln(n)}(x-2{)}^n.$

$(b)$ Consider the power series, $f(x)={\displaystyle \underset{n=0}{\overset{}{}}}{c}_n(x+3{)}^n.$ Suppose $we$ know that $(as$ a power series$)$

$f(-7),f^{\text{'}\text{'}}(11),$ and $f^{\text{'}}(5),$ diverge, while $(again,as$ series$)f^{\text{'}}(-3),f(1),$ and $f^{\text{'}\text{'}}(-5)$ converge.

Determine the radius $of$ convergence $of$ the power series for $f^{\text{'}\text{'}\text{'}}(x).$ Precisely name any result$(s)$

$(with$ the names from our lesson videos$)$ that you use, and where you are using them.