区間$[a,\,b]$を$N$分割した標本点を$x_k = a + hk$とおく、ただし、$h$は分割幅で $h=\frac{b-a}{N}$。ここで、$x_0=a$、$x_N=b$であることに注意。

また、 \begin{align} I_0&\equiv\int_a^b f(x)\mathrm{d}x=\sum_{k=0}^{N-1}\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x\\ I_n&\equiv\int_a^b f^{(n)}(x)\mathrm{d}x=f^{(n-1)}(b)-f^{(n-1)}(a) \end{align} と定義する。

$f(x)$を$x=x_k$の周りでテーラー展開したものを$[{x_{k}},\,{x_{k+1}}]$で積分し、和をとると、 \begin{align} f(x)&=f(x_k)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)(x-x_k)^n\\ \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x&=f(x_k)\left({x_{k}}-{x_{k+1}}\right)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)\int_{x_{k}}^{x_{k+1}}(x-x_k)^n\mathrm{d}x\\ &=hf(x_k)+\sum_{n=1}^{\infty}\frac{h^{n+1}}{(n+1)!}f^{(n)}(x_k)\\ I_0&=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^{n+1}}{(n+1)!}\sum_{k=0}^{N-1}f^{(n)}(x_k) \end{align} 同様に$f(x)$を$x=x_{k+1}$の周りでテーラー展開したものを$[{x_{k}},\,{x_{k+1}}]$で積分し、和をとると、

\begin{align} I_0&=h\sum_{k=0}^{N-1}f(x_{k+1})-\sum_{n=1}^{\infty}\frac{(-h)^{n+1}}{(n+1)!}\sum_{k=0}^{N-1}f^{(n)}(x_{k+1}) \end{align}

和をとる変数をそろえると、

\begin{align} I_0&=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\left(h\sum_{k=0}^{N-1}f^{(n)}(x_k)\right)\\ I_0&=h\sum_{k=1}^{N}f(x_{k})+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\left((-1)^{n}h\sum_{k=1}^{N}f^{(n)}(x_{k})\right) \end{align}

二式の平均をとると、$\frac{1}{2}\left(\right)$ \begin{align} I_0&=\frac{h}{2}\left(\sum_{k=0}^{N-1}f(x_k)+\sum_{k=1}^{N}f(x_{k})\right)\\ &+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)\right) \end{align}

$n=2m-1$のとき、 \begin{align} \sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k) &=\sum_{k=0}^{N-1}f^{(2m-1)}(x_k)-\sum_{k=1}^{N}f^{(2m-1)}(x_k)\\ &=f^{(2m-1)}(x_0)-f^{(2m-1)}(x_N) =-\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\\ &=-\int_a^b f^{(2m)}(x)\mathrm{d}x=-I_{2m} \end{align}

$n=2m$のとき、 \begin{align} \frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)\right) &=\frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(2m)}(x_k)+\sum_{k=1}^{N}f^{(2m)}(x_k)\right)\\ &=\frac{h}{2}\left(f^{(2m)}(x_0)+2\sum_{k=1}^{N-1}f^{(2m)}(x_k)+f^{(2m)}(x_N)\right)\\ &=h\left(\frac{f^{(2m)}(a)}{2}+\sum_{k=1}^{N-1}f^{(2m)}(x_k)+\frac{f^{(2m)}(b)}{2}\right)\\ &\equiv T_{2m} \end{align}

$$ I_0=T_{0} -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\frac{h}{2}I_{2m} +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2m} $$

$$ f(x)=f(x_{k+1})+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_{k+1})(x-x_{k+1})^n $$

$$ \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x=f(x_{k+1})h-\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_{k+1})(-h)^{n+1} $$

\begin{align} I(f)&=h\sum_{k=0}^{N-1}f(x_{k+1})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=0}^{N-1}f^{(n)}(x_{k+1})\\ &=h\sum_{k=1}^{N}f(x_{k})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=1}^{N}f^{(n)}(x_{k})\\ &=h\sum_{k=0}^{N-1}f(x_{k})-hf(x_0)+hf(x_N)+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}\left\{h\sum_{k=0}^{N-1}f^{(n)}(x_{k})-hf^{(n)}(x_0)+hf^{(n)}(x_N)\right\} \end{align}

\begin{align} I(f)&=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)})\\ I(f)&=R_h(f)+h\left(f(b)-f(a)\right)+\sum_{n=1}^{\infty}\frac{(-h)^n}{(n+1)!}\left\{R_h(f^{(n)})+h\left(f^{(n)}(b)-f^{(n)}(a)\right)\right\}\\ \end{align}

\begin{align} I(f)&=R_h(f)+\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}R_h(f^{(2m-1)})+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}R_h(f^{(2m)})\\ I(f)&=R_h(f)+h\left(f(b)-f(a)\right) -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{R_h(f^{(2m-1)})+h\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ &+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}\left\{R_h(f^{(2m)})+h\left(f^{(2m)}(b)-f^{(2m)}(a)\right)\right\}\\ \end{align}

\begin{align} I(f)&=R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right) -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{\frac{h}{2}\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ &+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}\left\{R_h(f^{(2m)})+\frac{h}{2}\left(f^{(2m)}(b)-f^{(2m)}(a)\right)\right\}\\ \end{align}

\begin{align} T_h(f)&\equiv R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right)\\ I(f^{(n)})&=f^{(n-1)}(b)-f^{(n-1)}(a) \end{align}

$$ I(f)=T_h(f) -\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})\\ $$

\begin{align} I(f)=T_h(f) -\frac{h^{2}}{2!}\frac{1}{2}I(f^{(2)})+\frac{h^{2}}{3!}T_h(f^{(2)}) -\frac{h^{4}}{4!}\frac{1}{2}I(f^{(4)})+\frac{h^{4}}{5!}T_h(f^{(4)}) -\frac{h^{6}}{6!}\frac{1}{2}I(f^{(6)})+\frac{h^{4}}{7!}T_h(f^{(6)})\\ +\cdots -\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) +\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})+\cdots\\ 0=\frac{h^{2}}{3!}I(f^{(2)})-\frac{h^{2}}{3!}T_h(f^{(2)}) +\frac{h^{2}}{3!}\frac{h^{2}}{2!}\frac{1}{2}I(f^{(4)})-\frac{h^{2}}{3!}\frac{h^{2}}{3!}T_h(f^{(4)}) +\frac{h^{2}}{3!}\frac{h^{4}}{4!}\frac{1}{2}I(f^{(6)})-\frac{h^{2}}{3!}\frac{h^{4}}{5!}T_h(f^{(6)})+\cdots&\\ 0=\frac{h^{2}}{3!}I(f^{(2)})-\frac{h^{2}}{3!}T_h(f^{(2)}) +\frac{h^{4}}{4!}I(f^{(4)})-\frac{h^{4}}{3!^2}T_h(f^{(4)}) +\frac{h^{6}}{3!4!2}I(f^{(6)})-\frac{h^{6}}{3!5!}T_h(f^{(6)})+\cdots&\\ \end{align} \begin{align} I(f)=T_h(f) -\frac{h^{2}}{2!}\frac{1}{6}I(f^{(2)}) +\frac{h^{4}}{4!}\frac{1}{2}I(f^{(4)}) +\frac{h^{4}}{3!}(\frac{1}{20}-\frac{1}{6})T_h(f^{(4)}) +\frac{h^{6}}{3!4!2}I(f^{(6)})-\frac{h^{6}}{6!}\frac{1}{2}I(f^{(6)}) +\frac{h^{4}}{7!}T_h(f^{(6)})-\frac{h^{6}}{3!5!}T_h(f^{(6)})+\cdots\\ +\cdots -\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) +\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})+\cdots\\ \end{align}