= Integral bounds for a finite harmonic sum
{title2=$\ln((N+1)/(m+1))<\sum_{n=m+1}^N1/n<\ln(N/m)$}
For integers $N>m\geq1$, 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 $n=m+1,\ldots,N$ 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.
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