ln1+2ln2+3ln3+...+nln(n) 有限项的和

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https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

$$\sum _{i=m}^{n}f(i)-\int _{m}^{n}f(x)~{\rm {d}}x={\frac {f(m)+f(n)}{2}}+{\frac {1}{6}}{\frac {f'(n)-f'(m)}{2!}}-{\frac {1}{30}}{\frac {f'''(n)-f'''(m)}{4!}}+\cdots +B_{2k}{\frac {f^{(2k-1)}(n)-f^{(2k-1)}(m)}{(2k)!}}+R_{2k}.$$