I Evaluating Taylor Series at the Mid-Point

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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:
$$f(x+\epsilon n, x) = f(x,x) + \epsilon n^{\mu}\frac{\partial f(x+\epsilon n,x)}{\partial n^{\mu}}\vert_{x+\frac{\epsilon}{2}n} + \mathcal{O}(\epsilon^{3}) $$.
 
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thatboi said:
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?
 
renormalize said:
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.
 
thatboi said:
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.
 
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