The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.

It is easy to show that

$$\sum_{1 \leq k } (\frac{1}{k} \times \frac{1}{k+1}) = 1.$$

The product $\frac{1}{k} \times \frac{1}{k+1}$ is equal to the area of a $\frac{1}{k}$by$\frac{1}{k+1}$ rectangle. The sum of the areas of these rectangles is equal to 1, which is the ares of a unit square. Can we use these rectangles to cover a unit square?

Is this problem still open?

What are the best results we know about this problem (or its relaxations)?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.

It is easy to show that

$$\sum_{1 \leq k } (\frac{1}{k} \times \frac{1}{k+1}) = 1.$$

The product $\frac{1}{k} \times \frac{1}{k+1}$ is equal to the area of a $\frac{1}{k}$by$\frac{1}{k+1}$ rectangle. The sum of the areas of these rectangles is equal to 1, which is the ares of a unit square. Can we use these rectangles to cover a unit square?

Is this problem still open?

What are the best results about this (or its relaxations)?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.

It is easy to show that

$$\sum_{1 \leq k } (\frac{1}{k} \times \frac{1}{k+1}) = 1.$$

The product $\frac{1}{k} \times \frac{1}{k+1}$ is equal to the area of a $\frac{1}{k}$by$\frac{1}{k+1}$ rectangle. The sum of the areas of these rectangles is equal to 1, which is the ares of a unit square.

Can we use these rectangles to cover a unit square?