One of the beautiful results in mathematics is the proof of the divergence of the harmonic series. What it tells us is that the infinite series of fractions
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …
gets infinitely large. Recently, I got to wondering which numbers it hits on the way up. In particular, if you can rearrange the fractions, can you hit any rational number?
I was thinking about this because I’ve been looking into the Egyptian fraction problem lately. The problem is a great one for students if you need to get a sense of how fractions really work. Unlike us moderns who would write 3/5 as the answer to how to divide 3 loaves of bread into five pieces, the Egyptians would first cut every loaf in half and give everyone a half, then divide the remaining half into five pieces. Their final instructions for the division would be that everyone gets 1/2 + 1/10.
Thus the problem of Egyptian fractions: given a fraction, can you always rewrite it as the sum of distinct unitary fractions, that is, fractions with a 1 in the numerator? (The Egyptians apparently didn’t like to repeat their fractions.) The next question is, how many unitary fractions does it take. This latter question is still unsolved in many cases. For example, it is conjectured that any fraction of the form 4/n can be written as the sum of at most three unitary fractions. But whether that’s always true is still unknown.
So here’s the harmonic puzzle: given any positive rational number, can you always write it as a sum of distinct unitary fractions?
When I first thought of this problem, it seemed like it would take very sophisticated tools to solve. Yesterday, I stumbled on the answer when working with a student on Egyptian fractions, and it takes nothing more sophisticated than a little algebra—and a clever idea.
Here, if you like, are the questions:
1. Can you write any rational number between 0 and 1 as the sum of distinct unitary fractions?
(Example: 4/13 = 1/5 + 1/10 + 1/130.)
2. (The Harmonic Puzzle) Can you write any positive rational number as the sum of distinct unitary fractions?
(Example: 2 = 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + +1/8 + 1/12 + 1/13 + 1/20 + 1/42 + 1/43 + 1/56 + 1/132 + 1/1806
Unless I made an arithmetic error… tell me if I did.)
3. (unsolved) Can you always write 4/n as the sum of three unitary fractions?
One thing that’s supercool: if you can answer question 2 in the affirmative, you get a slick proof that the harmonic series diverges!