Packing $1.35\cdot {10}^{11}$ rectangles into a unit square

It is known that ${\sum }_{i=1}^{\mathrm{\infty }}\frac{1}{i(i+1)}=1$. In 1968, Meir and Moser asked for finding the smallest $\epsilon$ such that all the rectangles of sizes $1\mathrm{/}i\times 1\mathrm{/}(i+1)$ for $i=1,2,\dots$, can be packed into a unit square or a rectangle of area $1+\epsilon$. In this paper, we show that we can pack the first $1.35\cdot {10}^{11}$ rectangles into the unit square and give an estimate for $\epsilon$ from this packing.