Integral comparison for a nonincreasing nonnegative function (${\int }_1^{N+1}f(x)dx\le {\sum }_{n=1}^{N}f(n)\le f(1)+{\int }_1^{N}f(x)dx$)

For a nonnegative nonincreasing function $f:[1,\infty )\to [0,\infty )$, comparison with unit-width rectangles givesOn $[n,n+1]$, the function is at most $f(n)$; on $[n-1,n]$, it is at least $f(n)$. Summing these inequalities gives the bounds. Applying them to $f(x)=1/x$ gives $\log (N+1)\le H_N\le 1+\log N$ for the harmonic numbers. Unlike a convergence verdict alone, the finite bounds estimate the size of a partial sum.