Consider the set of rectangles $r_n | n \in \Bbb N$ such that rectangle $r_n$ has shape $\frac1n\times\frac1{n+1}$. The total area composed by one copy of each $r_n$ as $n$ ranges from $1$ to infinity is $1$. Call that set of rectangles $S$.
In Concrete Mathematics it is speculated that you can fit all the rectangles in $S$ into a unit square, without overlap of any interior area. It is also speculated (by the second author) that they can't all be fit, and it is presented as a research problem.
I have a computer-based approach which could potentially decide the issue if you can't fit them (but won't provide a proof if they can fit). In particular, it seems to indicate that if you add the restriction that the rectangles (ordered by area) are forced to alternate long-side-vertical and long-side-horizontal, you come to an impasse.
But since the volume I saw it in is 25 years old, I'm wondering, before implementing and error-checking and optimizing the method for arbitrary placement, whether this question has, in the intervening years, been resolved.
So my question is
Is it known whether $S$ can be packed into a unit square, without overlap?