$(1)$ Consider the function $f(z)=\frac{1}{5-z}.$

$($a$)$ Find the Taylor expansion of $f(z)$ about the point $z_0=3,$ i$.$e$.,$ write $f(z)$ as $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n(z-3{)}^{\text{'}}$ and find the values of each $a_n.$

Hint: Recall from Calculus that the Taylor series has the form $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{f^{(n)}(a)}{n!}(z-a{)}^n.$

$(1)$ Consider the function $f(z)=\frac{1}{5-z}.$

$($a$)$ Find the Taylor expansion of $f(z)$ about the point $z_0=3,$ i$.$e$.,$ write $f(z)$ as $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n(z-3{)}^{\text{'}}$ and find the values of each $a_n.$

Hint: Recall from Calculus that the Taylor series has the form $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{f^{(n)}(a)}{n!}(z-a{)}^n.$

$(1)$ Consider the function $f(z)=\frac{1}{5-z}.$

$($a$)$ Find the Taylor expansion of $f(z)$ about the point $z_0=3,$ i$.$e$.,$ write $f(z)$ as $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n(z-3{)}^{\text{'}}$ and find the values of each $a_n.$

Hint: Recall from Calculus that the Taylor series has the form $f(z)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{f^{(n)}(a)}{n!}(z-a{)}^n.$