Harmonic Series

Definition of the Harmonic Series

A harmonic series is a specific type of infinite series where each term is the reciprocal of a positive integer. The harmonic series is given by:

S = \sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots

Each term $a_{n} = \frac{1}{n}$ decreases as $n$ increases, but the terms decrease very slowly compared to other series.

Example:

The sum of the first few terms of the harmonic series is:

S_{4} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = 2.0833

Even though the terms get smaller, the sum keeps increasing as more terms are added.

Divergence of the Harmonic Series

Despite the fact that the terms of the harmonic series get smaller and smaller, the harmonic series diverges. This means that as you keep adding more and more terms, the sum grows without bound.

Proof of Divergence:

One way to show that the harmonic series diverges is by grouping terms and comparing them to a series that grows without bound.

Consider:

S = 1 + \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) + (\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}) + \dots

Group the terms as follows:

The first group has 1 term: $1$ .
The second group has 1 term: $\frac{1}{2}$ .
The third group has 2 terms: $\frac{1}{3} + \frac{1}{4}$ , and each term is greater than or equal to $\frac{1}{4}$ .
The fourth group has 4 terms: $\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}$ , and each term is greater than or equal to $\frac{1}{8}$ .

For each group of terms, the total contribution is greater than or equal to $\frac{1}{2}$ . Since there are infinitely many such groups, the total sum of the series grows without bound, proving that the harmonic series diverges.

Thus, we conclude:

\sum_{n = 1}^{\infty} \frac{1}{n} = \infty

Comparison with $p$ -Series

The harmonic series is a special case of the $p$ -series, which has the general form:

\sum_{n = 1}^{\infty} \frac{1}{n^{p}}

Where $p$ is a constant.

Convergence and Divergence of $p$ -Series:

The $p$ -series converges if $p > 1$ .
The $p$ -series diverges if $p \leq 1$ .

The harmonic series is the case when $p = 1$ , and as we’ve seen, it diverges:

\sum_{n = 1}^{\infty} \frac{1}{n} (diverges)

Comparison Example:

For $p = 2$ , we have the convergent series:

\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots

This series converges to a finite value, specifically $\frac{π^{2}}{6}$ (a result from Euler).

For $p = \frac{1}{2}$ , we have the divergent series:

\sum_{n = 1}^{\infty} \frac{1}{n^{\frac{1}{2}}} = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots

This series diverges because $p \leq 1$ .