It is well known that the sum of a harmonic series does not have a closed form. Here is a formula which gives us a good approximation.
We need to find the sum of the following series
Consider the function , we intend to take middle riemann sums with rectangles of width starting from to .
Each rectangle in the figure has a width . The height of the rectangle is . The sum of the area of the rectangles is approximately equal to the area under the curve.
Area under f(x) from to
Apologies for the shabby graph.