\[\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\,\mathrm{d}x+\tfrac{1}{2}f(a)+\tfrac{1}{2}%
f(n)+\sum_{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^{(2s-1)}(n)-f^{(2s-1)}(a)%
\right)+\int_{a}^{n}\frac{B_{2m}-\widetilde{B}_{2m}\left(x\right)}{(2m)!}f^{(2%
m)}(x)\,\mathrm{d}x.\]