the harmonic series is just a discrete approximation of that integral

Answer by Mark Eichenlaub:

It doesn't, but it grows asymptotically as [math]\log n[/math] because[math]\log n \equiv \int_1^n \frac{1}{x} \mathrm{d}x[/math]and the harmonic series is just a discrete approximation of that integral.Please see my answer at Mark Eichenlaub's answer to Why is [math]\displaystyle\int \frac{\mathrm{d}x}{x} = \ln x[/math] ? for more.

What is an intuitive way to see why harmonic series sums to logn?

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