Consider the following equation
$$x^2-2(1+e)x+1-e=0$$
As part of an H.W assignment I'm requested to attempt and find the roots using perturbation theory, and compare to the exact solution.
When I substitute to $x(e)=\sum_{i=1}^n e^na_n$ and take up to first order in $e$ I obtain $$e^0(a_0-1)^2+e^1(2a_0a_1-2(a_0+a_1)-1)$$ by comparing the coefficients of $e$ to zero I get $$a_0=1$$ $$a_1=\frac{1+2a_0}{2(a_0-1)}$$ which immediately shows that the series converges.
I'm trying to understand why the perturbation method does not work.
When I look at the roots of the exact solution, they seems quite close to the zero'th order solution ($x=a_0=1$)
I'm quite sure that the reason it pointed out in the proximity between the roots, but can not explain it in mathematical terms.
Can anyone please advise?