We are told to evaluate the Taylor series of $f(z) = \dfrac{z+i}{z-3}$ around $z = 4i$ and that the final answer must be in summation notation
My work: (I have skipped steps for simplicity)
$f(z) = \dfrac{(z-4i)+5i}{(z-4i)-3+4i}$
$ = \dfrac{\sum_{n=0}^{\infty} \dfrac{z^{{n}}}{(4i)^n} + 20}{\sum_{n=0}^{\infty} \dfrac{z^{{n}}}{(4i)^n}+12i + 16}$
Even if i expand out the summation and add the values, I can't seem to get the final answer in summation notation. Any help would be appreciated!
$$f(z) = \frac{z+i}{(z - 4i) - 3 + 4i} = \frac{z+i}{(4i - 3)\left(1 + \frac{z - 4i}{4i-3}\right)}$$
Which will guarantee you the use of Geometric Series
$$\frac{1}{\left(1 + \frac{z - 4i}{4i-3}\right)} = \sum_{k = 0}^{+\infty}(-1)^k\left(\frac{z - 4i}{4i-3}\right)^k$$
Hence
$$f(z) = \frac{z+i}{4i-3}\sum_{k = 0}^{+\infty}(-1)^k\left(\frac{z - 4i}{4i-3}\right)^k$$