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Basel problem

The Basel problem asks for the exact value of the sum

 


Explanation

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 applies

so that

 


Additional information

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

 


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