Question

Given that $z$ is a non zero complex number, $i z ^2=1+\frac{2}{ z }+\frac{3}{ z ^2}+\frac{4}{ z ^3}+\frac{5}{ z ^4}+\ldots \ldots .+\infty$ and $z = n \pm \sqrt{-i}$ find the value of $100 n$.

Answer 1

Let $ S=1+\frac{2}{z}+\frac{3}{z^2}+\frac{4}{z^3}+\ldots \ldots$

$S =\frac{ z ^2}{( z -1)^2} $
$\therefore i z ^2 =\frac{ z ^2}{( z -1)^2} ( z \neq 0)$
$i =\frac{1}{( z -1)^2} \Rightarrow ( z -1)^2=\frac{1}{i}=-i \Rightarrow z -1= \pm \sqrt{-i}$
$z =1 \pm \sqrt{-i}$
$\text { given }z = n \pm \sqrt{-i} $
$\Rightarrow 100 n =100$