Saturday, 2 September 2017

real analysis - Show that suminftyn=1frac1n2=fracpi26




Show that n=11n2=π26 Anyone can help ?


Answer



This is known as the Basel problem and was first solved by Euler. His derivation (shown at above link) does some clever manipulations with the power series expansion of sin(x)x.




A more advanced proof uses Fourier transforms and Parseval's identity for the function f(x)=x.



The link also gives a rigorous but elementary proof, requiring only trigonometric identities and binomial coefficients, together with the "pinching lemma", but no calculus.


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