Euler–Maclaurin formula

Remarks

The proof is repeated integration by parts on each interval $[i, i + 1]$, using $B_k'(t) = k B_{k-1}(t)$ and the fact that $B_k(0) = B_k(1)$ for $k \ge 2$. For $p = 1$ the formula reads $\sum_{i=m}^{n} f(i) = \int_m^n f + \frac{f(m) + f(n)}{2} + \int_m^n f'(x) \bigl(x - \floor{x} - \frac12\bigr)\, dx$. Example: for $f(x) = x^2$ on $[0, n]$ with $p = 3$, the remainder vanishes since $f''' = 0$, and the formula gives $\sum_{i=0}^{n} i^2 = \frac{n^3}{3} + \frac{n^2}{2} + \frac{1}{12} \cdot 2n = \frac{n(n+1)(2n+1)}{6}$. The infinite series obtained by letting $p \to \infty$ usually diverges, so the formula is used with a fixed $p$ and an estimate of $R_p$.

Going deeper

For a student

In one direction it estimates sums by integrals: it gives Stirling's formula for $\ln n!$, asymptotics of harmonic numbers, and the values of power sums $\sum i^k$ as polynomials in $n$. In the other direction it estimates integrals by sums: it describes the error of the trapezoidal rule, and it explains why that rule is very accurate for smooth periodic functions. It is a quantitative link between series and the Riemann integral.