Mock AIME 2 2006-2007/Problem 5

Problem

Multiplying both sides of the equation by $z$ , we get

$iz^3 = z + 2 + \frac{3}{z} + \frac{4}{z^2} + \cdots$ ,

and subtracting the original equation from this one we get

$iz^2(z-1)=z+1+\frac{1}{z}+\frac{1}{z^2}+\frac{1}{z^3}+\cdots$ .

Using the formula for an infinite geometric series, we find

$iz^2(z-1)=\frac{z}{1-\frac{1}{z}}=\frac{z^2}{z-1}$ .

Rearranging, we get

$iz^2(z-1)^2=z^2\iff (z-1)^2=\frac{1}{i}=-i\Rightarrow z=1\pm\sqrt{-i}$ .