On packing of rectangles in a rectangle

It is known that ∑i=1∞1∕(i(i+1))=1. In 1968, Meir and Moser (1968) asked for finding the smallest ϵ such that all the rectangles of sizes 1∕i×1∕(i+1), i∈{1,2,…}, can be packed into a square or a rectangle of area 1+ϵ. First we show that in Paulhus (1997), the key lemma, as a statement, in the proof of the smallest published upper bound of the minimum area is false, then we prove a different new upper bound. We show that ϵ≤1.26⋅10−9 if the rectangles are packed into a square and ϵ≤6.878⋅10−10 if the rectangles are packed into a rectangle.