Basel problem

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The Basel problem asks for the exact value of the sum

We start with the power series for the sine and divide through by x and obtain

The zeroes of this function are integer multiples of π, so the points x = n · π for n = ±1, ±2, ±3, ... . The factorisation theorem gives

what you can convert to

The multiplication of these factors show a clear pattern

We only continue with the coefficients of x² and find

In the original series we can see that the coefficient of x² equals there

and so holds

so that

The first calculation of this probem from 1644 was given by Leonhard Euler in 1735.