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

Mark's answer is good. But you asked for a way to "see" why
$\sum_{k=1}^{n} \frac{1}{k} \sim \log{n}$
Below I have plotted the function $f(x) =1/x$ in red along with a series of rectangles, each one corresponding to a term in the sum. The first big rectangle on the left corresponds to the $k=1$ term with an area of 1, the second smaller rectangle corresponds to the $k=2$ term with an area of 1/2, and so on. You can see that the rectangles of increasing $k$ get closer and closer to the curve $f(x) =1/x$.
Now, the sum is just the total area of all of the $n$ rectangles. If $n$ is very large, then this area is approximately equal to the area under the red curve from $x=1$ to $x=n$. But the area under the curve is just the integral of $f(x),$
$\int_1^n f(x) dx =$$\int_1^n \frac{dx}{x} = \log{n}$
Moreover, as $n$ increases this approximation gets better, because the rectangles and the curve get closer as $k$ increases. So, we say that the two areas are asymptotically equal as $n$ goes to infinity.

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