# Alternating Harmonic Series

### Insight into a peculiar series that approximates an integral

Hi Scholars!

Many series converge to rather counterintuitive constants. The alternating harmonic series is one example of this, and this graphic shows how the series can be rearranged to obtain an integral approximation that converges to the logarithm of 2.

It took me a few minutes of playing around to see why this was, in fact, the case. Coming up with a way to visualize a series was a difficult challenge. The shifting of rectangles with the changing expression for the sum helped me better understand why terms cancel out graphically. Notice that the height and width of the rectangles changes once we move to an integral approximation. Rather than filling up from 1 to n, it goes from 1 to 2, which is why the width of the rectangles starts getting so small. Anyway, this was a fun one. Feel free to suggest other similar problems to try and visualize!

Stay Awesome.

Howard Heaton