## The Harmonic Series and Beyond

I was going to write a post about AQUA@Home, the cool quantum simulation system you can help with that I’ve been working on at D-Wave, but I’ll leave that for a couple of days and talk about one of the miscellaneous things that I’ve thought about at random.

I’ve known for quite some time that (the harmonic series) divergesas n approaches infinity, and more recently, that it scales as . However, converges, along with any other positive function being summed. For a while I had assumed that any function would converge, but I learned in my second calculus course at Carleton that also diverges.

Several weeks ago, I was curious where the true bound between converging and diverging is, and I seem to have found it.

I started by finding a simple proof of the scaling of the harmonic series, and it goes something like the following. Let’s assume that n is a power of two:

and:

Therefore, . Using a similar approach, one can show that . Doing this again can show that , and a pattern emerges allowing each one to be expressed in terms of the previous.

The final conclusion:

where the number of logs in the last factor and in the scaling is *k*. Note that the is just to prevent any of the logarithms from going to zero, so assume . If any one of the logarithmic factors in the sum has a power greater than one, the series converges, but with them all equal to or less than one, the series still diverges. In effect, the scaling can get arbitrarily close to , so this appears to be the very brink of divergence. Some may argue that having a finite *k* makes it insufficient, since there are theoretically slower-growing functions, but that’s really splitting hairs.