I have an equation: $(1-\epsilon)x^2 -2x +1=0$ (regularly perturbed problem, we anticipate all roots to remain bounded when $\epsilon$ goes to $0$)
I substitute $x=\displaystyle \sum_{n\geq0}C_n \epsilon^n$ and have to work out the equations for $C_0,C_1,C_2$.
So I get: $C_0^2-2C_0+1=0$ so $C_0=1$
and there a problem starts: as the equation for $C_1$ is: $2C_0C_1-C_0^2-2C_1=0$ so we get $1=0$. What is the reason this approach fails and how to devise a way to fix it?