## Harmonic Numbers

The $n$th harmonic number $h(n)$ is defined as $$h(n) = \sum_{i=1}^n 1/i$$ The harmonic series is the associated series $\sum_{i=1}^\infty 1/i$ and it diverges. There are probably quite a few interesting ways to see this. My favourite is a simple comparison test: $$1/1 + 1/2 + 1/3 + \cdots\\ \geq 1/2 + 1/2 + 1/4 […]