I am learning perturbation theory. I found this equation from online for perturbation theory
$$ x = X_0 + εX_1 + ε^2 X_2 + O(ε^3) \tag 1$$
The square of equation $(1)$ is $$x^2 = X_0^2 + 2εX_0 X_1 + ε^2 (X_1^2 + 2X_0X_2) + O(ε^3) \tag 2$$
Equation (2) is not clear to me. If it is follows $(a+b)^2$ formula how the third term comes? Can anyone explain which formula followed for Eq. $(2)$? Thanks.
It's exactly as you say, a consequence of $(a+b)^2=a^2+2ab+b^2$. Indeed,
$$(X_0 + εX_1 + ε^2 X_2 + O(ε^3))^2=X_0^2 + 2X_0(εX_1 + ε^2 X_2 + O(ε^3))+(εX_1 + ε^2 X_2 + O(ε^3))^2$$
So, you already have the first two terms and part of the third term:
$$X_0^2 + 2εX_0X_1 + 2ε^2 X_0X_2 + O(ε^3)+(εX_1 + ε^2 X_2 + O(ε^3))^2$$
Then squaring the last term
$$(εX_1 + ε^2 X_2 + O(ε^3))^2 = ε^2X_1^2 + 2εX_1(ε^2 X_2 + O(ε^3)) + (ε^2 X_2 + O(ε^3))^2$$
You obtain the missing piece of the third term. All remaining terms are $O(\epsilon^3)$.