Another way of showing the harmonic series diverges, without all those fancy comparisons to integrals (and having to know that lim_{x->infinity} log(`x`) = infinity, which is somewhat harder to prove), is to perform the following elementary comparison.

1 >= 1
1/2 >= 1/2
1/3 >= 1/4
1/4 >= 1/4
1/5 >= 1/8
1/6 >= 1/8
1/7 >= 1/8
1/8 >= 1/8
...

Generally,

*each* of the terms 1/(2

^{k}+1), 1/(2

^{k}+2), ...,

1/2

^{k+1} is

at least 1/2

^{k+1}, and there are 2

^{k} such terms. Thus, the sum of these terms is

*at least* 1/2, so the sum of the first 2

^{k} terms of the harmonic series is at least

`k`/2; thus, the

series diverges.

Well actually, I lied. This sort of thing is called the condensation test for series convergence...