Harmonic series

Here we will look at the harmonic series. To investigate its convergence and perhaps determining its sum we need to investigate the sequence of its partial sums:

This sequence can be also defined recursively:
(1) s₁ = 1,
(2) s_n = s_n−1 + 1/n, for n = 2,3,4,...

Since every term is obtained by adding a positive number to the previous term, this sequence is increasing. Thus there are two possibilities. Either the sequence is bounded and converges, or it is unbounded from above and goes to infinity. Which is the right one?

It is easy to write a simple program for a computer or a programmable calculator to calculate the n-th term of this sequence for us. We then obtain for instance this:

s₁₀₀ = 5.18738,
s₁₀₀₀ = 7.48548,
s₁₀₀₀₀ = 9.78761.

What happens if we try the trick with successive numbers? For instance, s_1000000 = 14.35736 (six zeros in the index) and s_1001000 = 14.35831. One would guess that the sequence converges and the limit is most likely about 14.36. However, it is not difficult to show that actually s_n > ln(n), therefore this sequence goes to infinity.

If you feel that this example is too simple and that you would see through it, take out your calculator and try to put some numbers into a_n = ln(ln(ln(ln(n)))). For instance, a_1000000000 = 0.1034383 (nine zeros in the index), and if we double the index, we get a_2000000000 = 0.1131257. The largest number in my calculator is 10 to the power 99. If I use it as n, I get a_n = 0.52578301. Wouldn't you say that this sequence converges, probably never exceeding 1? In fact it also goes to infinity.