Evaluating Taylor Series at the Mid-Point

[thatboi]

Hi all,
I came across the following stackexchange post and was wondering if anyone could give any elaboration for why the answer claims that evaluating the Taylor Series resulted in $\mathcal{O}(\epsilon^{3})$ errors? I have not encountered such an expansion before.
EDIT: The equation at hand is:
$$f(x+\epsilon n, x) = f(x,x) + \epsilon n^{\mu}\frac{\partial f(x+\epsilon n,x)}{\partial n^{\mu}}\vert_{x+\frac{\epsilon}{2}n} + \mathcal{O}(\epsilon^{3}) $$.

 

[renormalize]

Hi all,
I came across the following stackexchange post and was wondering if anyone could give any elaboration for why the answer claims that evaluating the Taylor Series resulted in $\mathcal{O}(\epsilon^{3})$ errors? I have not encountered such an expansion before.

To keep PF threads as self-contained as possible, could you post a summary here of the answer you're asking about?

 

[thatboi]

To keep PF threads as self-contained as possible, could you post a summary here of the answer you're asking about?

Ah that is true. I have edited the question.

 

[renormalize]

Ah that is true. I have edited the question.

Thanks. To see why the posted equation has errors of $\mathcal{O}(\epsilon^{3})$ rather than $\mathcal{O}(\epsilon^{2})$, take a look at: https://en.wikipedia.org/wiki/Midpoint_method.