# Taylor series expansion about point $x_0 + \epsilon$

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

• They probably meant " The Taylor Series expanded about $x_0$ to evaluate small variations from $x_0$, $x=x_0+\epsilon$ ". Nov 18 '20 at 2:45
• so the taylor expansion is about point $x_0$ and we use the function approximation at point $x_0$ to estimate the value at $x_0 + \epsilon$ ? Nov 18 '20 at 2:46
• Exactly. That way you obtain a function for $\epsilon$: since $x_0$ is a constant, the only variable in the expression of $f(x_0+\epsilon)$ is $\epsilon$ itself. Nov 18 '20 at 2:52
• so this function of $\epsilon$ allows me to obtain the value of $f(x_0 + \epsilon)$ for any small values of $\epsilon$ ? Nov 18 '20 at 3:03
• Yes. In this context, another common notation is to write $\epsilon$ as $\Delta x$: $f(x_0+\Delta x) \equiv f(\Delta x) = f(x_0) + f'(x_0) \Delta x$ Nov 18 '20 at 3:09

It's kind of a weird mixed series obtained from the Taylor series. Start with the Taylor series about $$x=x_0$$, namely
$$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+ \dots$$
Now substitute $$x-x_0=\varepsilon$$ on the RHS, but leave the evaluations of $$f$$ and its derivatives on the RHS at $$x_0$$. Then on the LHS, write $$f(x)=f(x_0+\varepsilon)$$. This gives
$$f(x_0+\varepsilon)=f(x_0)+f'(x_0)\varepsilon+\frac{1}{2}f''(x_0)\varepsilon^2+ \dots$$