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:
$$1/1 + 1/2 + 1/3 + \cdots\\ \geq 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + 1/8 + \cdots
\\= 1 + 1 + 1 + \cdots$$
and the series $1 + 1 + \cdots$ is divergent. But while the harmonic series diverges, it does so rather slowly. It does so slowly enough that if you were to numerically compute the harmonic numbers (the partial sums of the harmonic series), you might be unconvinced that it actually does diverge:

  • $h(10) = 2.92896825396825\dots$
  • $h(100) = 5.18737751763962\dots$
  • $h(1000) = 7.48547086055035\dots$
  • $h(10000) = 9.78760603604438\dots$
  • $h(100000) = 12.0901461298634\dots$
  • $h(1000000) = 14.3927267228657\dots$

These numbers were computed by actually summing $1 + 1/2 + \cdots + 1/n$ and then writing out a decimal approximation to that fraction, but that takes a while. How can we at least give an approximation to this series? The first thought surely must be to compare it to the integral
$$\int_1^n 1/x dx = \log(n)$$
Where $\log(-)$ denotes the natural logarithm. A moment's consideration with Riemann sums shows that we have an inequality
$$\int_1^n 1/x dx \le h(n) \le \int_1^n 1/x dx + 1$$
So we've come up with a pretty good approximation to our harmonic series, which only gets better as $n$ gets bigger:
$$\log(n)\le h(n) \le \log(n) + 1$$
Which, incidentally is another explanation of why the harmonic series diverges. And it's much faster to compute on a simple scientific calculator. Here is an example computation: we have already said that $h(1000000) = 14.3927267228657\dots$ But $\log(1000000) = 13.8155105579643\dots$ Pretty good right?

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