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m I explained why one may chose a shorter proof than Euler's one. |
m I justified why one can separate odd and even numbers. |
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The sum and the integral signs were interchanged thanks to Beppo Levi's [[monotone convergence theorem]] for Lebesgue integral. Then, [[Wallis' integrals]] enabled us to integrate the powers of sine.
One can separate even and odd numbers since all the terms are positive :
<math>\displaystyle \sum ^{+\infty }_{n=1}\frac{1}{n^{2}} =\sum ^{+\infty }_{n=1}\frac{1}{( 2n)^{2}} \ +\sum ^{+\infty }_{n=0}\frac{1}{( 2n+1)^{2}} \
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