A good ques in complex nos.

hint:infinite geometric series will be used in the soln

\texttt{Let } S=\frac{1}{z}+\frac{2}{z^{2}}+... \Longrightarrow \frac{S}{z}= \frac{1}{z^{2}}+\frac{2}{z^{2}}+.... \texttt{Subtracting term by term } S\left(\frac{z-1}{z}\right)=\frac{1}{z-1} \texttt{so, } S=\frac{z}{(z-1)^{2}}

You're right...rajat

Z=1-2i

S = 1/Z + 2/Z² + 3/Z³ + 4/Z⁴ + ...........

S.Z = 1 + 2/Z + 3/Z² + 4/Z³ + 5/Z⁴ + ...........

S(Z-1) = 1 + 1/Z + 1/Z² + 1/Z³ + 1/Z⁴ + ...........
= Z/(Z-1)

So, S = Z/(Z-1)² = (1-2i)/(-2i)² = -(1-2i)/4