Placing perturbation series in a function problem

Question

I have trouble under standing how to put perturbation series in a function.

In "eq-a" I would have assume $f(x_0) + \epsilon f'(x_0) + \epsilon^2 f''(x_0)$

In "eq-a"..how the term $ (\epsilon x_1 + \epsilon^2 x_2) \epsilon^2 f'(x_0) + \frac{1}{2} \epsilon^2 x_1^2 f''(x_0)$ came about ?

I will really appreciate the help.

Answer 1

With $x = x(\epsilon,\tau) = x_0 + \epsilon x_1 + \epsilon^2 x_2 + \ldots$, $$\eqalign{f(x ) &= f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2} (x - x_0)^2 + \ldots \cr &= f(x_0) + f'(x_0) (\epsilon x_1 + \epsilon^2 x_2 + \ldots) + \frac{f''(x_0)}{2} \left( \epsilon x_1 + \epsilon^2 x_2 + \ldots\right)^2 + \ldots\cr &= f(x_0) + \epsilon f'(x_0) x_1 + \epsilon^2 \left(f'(x_0) x_2 + \frac{f''(x_0)}{2} x_1^2\right) + \ldots}$$