The taylor series expansion about point $a$ for function $f(x)$ is given by $$f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2!}f''(a)(x-a)^2 +...$$
As more and more terms are added to the taylor polynomial, the interval of approximation over $f(x)$ increases as well.
For a taylor series expansion about point $x_0 + \epsilon$, shouldn't the expansion be $$f(x) \approx f(x_0 +\epsilon) + f'(x_0+\epsilon)(x-(x_0+\epsilon)) + ... $$
Why do they write it as $$f(x_0 + \epsilon) = f(x_0) + f'(x_0)\epsilon + \frac{1}{2}f''(x_0)\epsilon^2$$ This is taken from wolfram alpha