Harmonic number ($H_n$)

The $n$th harmonic number is $H_n={\sum }_{k=1}^{n}1/k$. Its asymptotic expansion beginswhere $\gamma$ is the Euler--Mascheroni constant.

For integers $N>m\ge 1$, compare the decreasing function $1/x$ with unit rectangles: on $n-1<x<n$, $1/(x+1)<1/n<1/x$. Integrating and summing over $n=m+1,\dots ,N$ proves the displayed strict bounds. Adding a known harmonic number $H_m$ gives bounds on $H_N$ without computing every omitted term. A requested rounding is certified when both bounds round to the same value; a midpoint estimate alone does not establish the rounding.