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The Euler-Maclaurin formula relates a finite sum to an integral and endpoint derivatives. If $a<b$ are integers, $p\ge 1$ and $f$ has continuous derivatives through order $2p$ on $[a,b]$, thenHere $B_{2r}$ are Bernoulli numbers. With ${\tilde{B}}_{2p}(x)=B_{2p}(\{x\})$ the periodic Bernoulli function, the exact remainder isThis form rearranges the last endpoint correction in the NIST statement.^[ref]

For $f(x)=x^2$ on $[0,n]$ and $p=1$, the remainder vanishes because the second Bernoulli polynomial has integral zero over each unit interval. The result is ${\sum }_{k=1}^n{k}^2=n^3/3+n^2/2+n/6$, the square pyramidal number formula. For other functions, retaining or bounding $R_p$ makes the finite truncation a controlled approximation.

Official text: dlmf.nist.gov/2.10

Section 2.10(i), equation 2.10.1 gives the Euler-Maclaurin formula with an exact periodic Bernoulli function remainder. Moving the constant Bernoulli-number term out of the final integral yields the endpoint-correction convention used here.

The periodic Bernoulli function is the periodic extension ${\tilde{B}}_n(x)=B_n(\{x\})$ of the Bernoulli polynomial, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part. It has period one. For example, ${\tilde{B}}_2(x)=\{x{\}}^2-\{x\}+1/6$.

For $n\ge 2$, its Fourier expansion isThe series is absolutely convergent. For $n=1$ the same formula holds away from the integers and describes a sawtooth. At an integer the symmetric Fourier value is zero, whereas the literal fractional-part convention gives $B_1(0)=-1/2$; endpoint conventions must therefore be distinguished.^[ref]

Periodic Bernoulli functions occur in the remainder of the Euler-Maclaurin formula, connecting errors in discrete summation with periodic oscillation between successive integers.