10.5 Harmonic Series and \(p\)-Series

Definition: \(p\)-Series Test

For the series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\):

1. If \(p > 1\), the series converges.

2. If \(p \le 1\), the series diverges.

Note: When \(p=1\), it is called the Harmonic Series.

Extension: The Basel Problem

The \(p\)-series test confirms that \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges since \(p=2 > 1\). However, finding the exact value was a major challenge until Leonhard Euler solved it in 1734.

The Discovery:

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots = \frac{\pi^2}{6}\]

How he did it (The Intuition):

Euler treated the Maclaurin series for \(\frac{\sin x}{x}\) as if it were an infinite polynomial:

\[\frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \dots\]

By comparing this to the roots of the sine function (which occur at \(x = \pm \pi, \pm 2\pi, \dots\)), he elegantly extracted the value \(\frac{\pi^2}{6}\). This result is also the value of the Riemann Zeta Function at \(s=2\), or \(\zeta(2)\).