オイラーの和公式
( f ( k − 1 ) ( n ) − f ( k − 1 ) ( 0 ) ) − R m {\displaystyle \int _{0}^{n}f(x)dx=\sum _{j=0}^{n-1}f(j)-{\frac {1}{2}}f(0)+{\frac {1}{2}}f(n)-\sum _{k=2}^{m}(-1)^{k}{\frac {B_{k}}{k!}}\left(f^{(k-1)}(n)-f^{(k-1)}(0)\right)-R_{m}}
R m = ( − 1 ) m + 1 ∫ 0 n B m ( x − ⌊ x ⌋ ) m ! f ( m ) ( x ) d x {\displaystyle R_{m}=(-1)^{m+1}\int _{0}^{n}{\frac {B_{m}(x-\lfloor {x}\rfloor )}{m!}}f^{(m)}(x)dx}
を得る。移項して形式を整えると
∑ j = 0 n − 1 f ( j ) = ∫ x = 0 n f ( x ) d x + ∑ k = 1 m B k k ! ( f ( k − 1 ) ( n ) − f ( k − 1 ) ( 0 ) ) + R m {\displaystyle \sum _{j=0}^{n-1}f(j)=\int _{x=0}^{n}f(x)dx+\sum _{k=1}^{m}{\frac {B_{k}}{k!}}\left(f^{(k-1)}(n)-f^{(k-1)}(0)\right)+R_{m}}