Hi

I read in a book that in a non linear circuit the relation between the current and voltage involves the power series like this :

I = a1.V + a2.V^2 + a3.V^3 + a4.V^4 ...

how is this relation derived for a circuit ?

 

how is this relation derived for a circuit ?

It entirely depends upon the nature of the non-linear circuit.
For example, a junction diode has a logarithmic relation between the voltage across it and the current through it.