= Solution
For a strictly decreasing function, $f(n+1)<f(x)<f(n)$ in the interior of $[n,n+1]$. Integrating over the unit interval and summing from $n=r$ to $s-1$ gives the <integral comparison for a nonincreasing nonnegative function>:
$$
\boxed{\sum_{n=r+1}^sf(n)<\int_r^sf(x)dx<\sum_{n=r}^{s-1}f(n).}
$$
Geometrically, right-endpoint rectangles lie below the curve and left-endpoint rectangles above it. If “decreasing” permits flat pieces, use non-strict inequalities unless strictness is ensured on some interval. When $r=1$, the endpoint value $f(1)$ is also needed; the power function below is defined there.
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