# Sequences and Series, The Harmonic Series

## The Harmonic Series

The series h_{n} = 1/n is called the harmonic series,
since the terms represent the harmonics of a fundamental wave form.
This is an example of a divergent series whose terms approach 0.
To prove this, we will show that h dominates an unbounded series.
Define the series g as:
1 1/2 1/4 1/4 1/8 1/8 1/8 1/8 1/16 (8 of these) 1/32 (16 of these) …

The second term is 1/2, and the third and fourth terms add up to 1/2.
The next four terms add up to 1/2,
the next 8 terms add up to 1/2, the next 16 terms add up to 1/2, and so on.
We keep adding halves forever, hence g is unbounded.
Since h dominates g, h is unbounded.

Let r be the sequence of partial sums of h.
Recall that the log of n is the area under the curve y = 1/x from 1 to n.
Now r_{n-1} is a set of regularly spaced rectangles approximating the area of the curve.
Shifting the rectangles to the left one unit, r_{n}-1 also approximates the area.
The former is too large (upper sums), while the latter is too small (lower sums).
We thus have
r_{n}-1 < log(n) < r_{n-1}.
Note that log(n) is bracketed between two numbers that are never more than 1 apart.
Thus the difference between r_{n} and log(n) is no more than 1.