Regular way to fill a 1×11\times1 square with 1n×1n+1\frac{1}{n}\times\frac{1}{n+1} rectangles?

The series n=11n(n+1)=1 suggests it might be possible to tile a 1×1 square with nonrepeated rectangles of the form 1n×1n+1. Is there a known regular way to do this? Just playing and not having any specific algorithm, I got as far as the picture below, which serves more to get a feel for what I am looking for.

Tiling of Square with rectangles

I think some theory about Egyptian fractions would help. It’s nice for instance in the center where 13+14+16+14=1. And on the right edge where 12+13+16=1.

Side note: The series is (1112)+(1213)+(1314)+. The similar looking (1112)+(1314)+(1516)+ sums to ln(2), and there is a nice picture for that, if you interpret ln(2) as an area under y=1x:

Tiling of ln(2) with rectangles


Source : Link , Question Author : alex.jordan , Answer Author : Community

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