The problem here is the sum,
We have no "nice" formula for working with .
However, in analysis we should learn how to approximate.
Here is how we do it.*
Define the following function (see below).
Now the integral:
Is the area below this curve.
In this particular case it is:
And the area of the rectangles is (the width is 1):
Thus, we have:
In general we have,
*)The credit goes to the proof of the Integral Test for series.