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

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.

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

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