Asymptotic of Riemann Sum I want to prove that $$\frac{b-a}{n}\sum_{k=1}^nf\left(a+k\frac{b-a}{n}\right)=\int_a^bf(t)dt+\frac{\alpha}{n}+\frac{\beta}{n^2}+o\left(\frac{1}{n^2}\right)$$
where $f\in\mathcal{C}^2([a,b],\mathbb{R})$  for some $\alpha$ and $\beta$
 A: $\newcommand{\+}{^{\dagger}}%
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$$
{b - a \over n}\sum_{k = 1}^{n}\fermi\pars{a + k\,{b - a \over n}}
={b - a \over n}\fermi\pars{b}
+ {b - a \over n}\sum_{k = 1}^{n - 1}\fermi\pars{a + k\,{b - a \over n}}
$$
With $\it\mbox{Euler-Maclaurin Summation Formula}$:
\begin{align}
&\sum_{k = 1}^{n - 1}\fermi\pars{a + k\,{b - a \over n}}
\\[3mm]&=
\int_{0}^{n}\fermi\pars{a + k\,{b - a \over n}}\,\dd k
-\half\bracks{\fermi\pars{a} + \fermi\pars{b}}
+
{1 \over 12}\bracks{\fermi'\pars{b} - \fermi'\pars{a}}{b - a \over n} + \cdots
\\[3mm]&=
{n \over b - a}\int_{a}^{b}\fermi\pars{x}\,\dd x
-\half\bracks{\fermi\pars{a} + \fermi\pars{b}}
+
{1 \over 12}\bracks{\fermi'\pars{b} - \fermi'\pars{a}}{b - a \over n} + \cdots
\end{align}
Can you complete it ?
