$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$
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THEOREM 2 Suppose that $\sum_{n=0}^\infty a_n(z — zo)^n$ is a power series with a positive or infinite radius of convergence $R$.
(a) \begin{equation} \frac{1}{R}= \lim_{n\to \infty} |\frac{a_{n+1}}{a_n}| \tag{6} \end{equation} provided the limit on the right exists.
(b) \begin{equation} \frac{1}{R}= \lim_{n\to \infty} \sqrt[n]{|a_n|} \tag{6} \end{equation} provided the limit on the right exists.
Find the radius of convergence: $$ \sum_{n=1}^\infty \frac{(2n)!!}{(2n-1)!!} z^n $$ where $(2n)!!= 2n (2n-2)(2n-4)\cdots 4\cdot 2$, $(2n-1)!!= (2n-1) (2n-3)(2n-5)\cdots 3\cdot 1$.
Find the power series about the origin for the given function. $$ \frac{z^2}{(4-z)^2},\qquad |z|< 4. $$ Hint: $(4-z)^{-2}=\frac{d}{dz}(4-z)^{-1}$.
Find a “closed form” (that is, a simple expression) for each of the given power series. $$ \sum_{n=0}^\infty n(n-1)z^n $$ Hint: divide by $z^2$.