It is known that $\sum\limits_{i=1}^{\infty} \frac{1}{i (i+1)} = 1$. In 1968, Meir and Moser asked for finding the smallest $ε$ such that all the rectangles of sizes $1/i \times 1/(i + 1)$ for $i = 1, 2, \ldots$, can be packed into a unit square or a rectangle of area $1 + ε$. In this paper, we show that we can pack the first $1.35\cdot10^{11}$ rectangles into the unit square and give an estimate for $ε$ from this packing.