The forms eulermac(expr, x) and eulermac(expr, x, n) compute asymptotic approximations to sum(expr, x). If F(x) = eulermac(f(x), x), then $F\left(x+1\right)-F\left(x\right)$ is asymptotically equivalent to f(x). The order of the approximation is specified by n, and defaults to Order - 1.

The forms eulermac(expr, x=a..b) and eulermac(expr, x=a..b, n) compute nth degree Euler-Maclaurin summation formulas for expr (thus n terms of the expansion are given). If n is not specified, it is assumed to be Order - 1.

$\mathrm{eulermac}\left(\frac{1}{x}\,x\right)$

$\ln \left(x\right)-\frac{1}{2x}-\frac{1}{12x^2}+\frac{1}{120x^4}-\frac{1}{252x^6}+\mathrm{O}\left(\frac{1}{x^8}\right)$

$\mathrm{eulermac}\left(\frac{1}{k}\,k=1..x\right)$

${\int }_1^x\frac{1}{k}\ⅆk+\mathrm{\gamma }+\frac{1}{2x}-\frac{1}{12x^2}+\frac{1}{120x^4}-\frac{1}{252x^6}+\mathrm{O}\left(\frac{1}{x^8}\right)$

$\mathrm{eulermac}\left(\frac{1}{x^2}\,x\right)$

$-\frac{1}{x}-\frac{1}{2x^2}-\frac{1}{6x^3}+\frac{1}{30x^5}-\frac{1}{42x^7}+\mathrm{O}\left(\frac{1}{x^9}\right)$

$\mathrm{eulermac}\left(\frac{1}{k^2}\,k=1..x\,4\right)$

${\int }_1^x\frac{1}{k^2}\ⅆk+\frac{{\mathrm{\pi }}^2}{6}-1+\frac{1}{2x^2}-\frac{1}{6x^3}+\frac{1}{30x^5}+\mathrm{O}\left(\frac{1}{x^7}\right)$