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Euler’s amazing identity

The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity:

Equation 1:##1 + 1/4 + 1/9 + 1/16 + … = \pi^2/6##

This is one instance of an important function called the Riemann Zeta function, ##\zeta(s)##, which in the case where ##s > 1## is defined by:

Equation 2:##\zeta(s) = \sum_{j=1}^\infty \dfrac{1}{j^s}##

So Euler’s identity can be written as:

Equation 3:##\zeta(2) = \frac{\pi^2}{6}##

This post is an attempt to show how you can derive that result, and related results, using facts about trigonometry, complex numbers, and the Fourier series.

Some related functions defined...

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