Asymptotic of Riemann Sum

Question

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 $\beta$

Answer 1

$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $$ {b - a \over n}\sum_{k = 1}^{n}\fermi\pars{a + k\,{b - a \over n}} ={b - a \over n}\fermi\pars{b} + {b - a \over n}\sum_{k = 1}^{n - 1}\fermi\pars{a + k\,{b - a \over n}} $$ 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 ?