harmonic series

If I add all the fractions

1+1/2+1/3+1/4+1/5+...

all the way to infinity, how far do I get?

The series is called the harmonic series because all the different harmonics of a string have these different wavelengths. So this series is almost like hearing all the harmonics of the vibrating string.

The question of how big this series is can be expressed physically too. If I have infinitely many books and I start stacking them on top of each other, how far can I get them to overhang? In the picture above I've used CD cases (Dorothy will be more interested in which CDs are being stacked probably - there was some Schubert, Shostokovich, Puccini, Stanley Turrentine, Elvis Costello, Dvorak and My Bloody Valentine.) In the picture, the top CD is completely clear of the CD at the base of the tower. As one descends the tower, the distance that each CD can be pushed so that the thing still balances is given by the harmonic series. So the reason the first CD is clear of the base is because the overhang of 4 CDs is 1/2x(1+1/2+1/3+1/4) which is bigger than 1. But how far can you go if you keep stacking the CDs or books?

Well, it turns out that I can actually get as far as I want. This is because

1+1/2+1/3+1/4+1/5+... gets infinitely big.

The first proof of this was given by the fourteenth century philosopher Nicole d'Oresme.

The proof is not too difficult but clever. I might give it in another blog entry if there is popular demand!

How far can we get with 19 steps: the top step will be 1.77387... units away from the step at the bottom. Although the series goes off to infinity, it takes a very long time to get there. How many steps do we need for the top step to be 2 units away? That takes 31 steps. And to go 5 units...that requires 12367 steps.

This paper explains some of the maths behind stacking books.