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Find the sum $$\sum _{n=1}^{\infty} {1\over n^2}$$ elementary! I know the result is $\pi^2/6$, but I wonder if there is an elementary approach i.e. with a high school knowledge to this sum.

nonuser
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    Elementary? Fine: $$\sum_{n=1}^\infty\frac1{n^2}=\zeta(2)=\frac\pi6$$ Elementary is in the eyes of the beholder...elementary, Watson! And I doubt there are many high school in the globe that teach, as usual curriculum, the needed stuff to understand even the most elementary proofs of this fact. – DonAntonio Dec 24 '17 at 21:41
  • @ DonAntonio Ok, but it is not $\pi^2/6$? So the answer is negative? – nonuser Dec 24 '17 at 21:47
  • Your first try was $6/\pi$, remember? –  Dec 24 '17 at 21:49
  • There is an elementary way, since it was thesingle problem for the French ‘baccalauréat’ (high school final exam) some 30 years ago. Elementary, but long,since highschool student do not know much mathematics. They were allotted 4 hours for this exam problem. – Bernard Dec 24 '17 at 21:49
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    Can Euler's approach be considered elementary? https://en.wikipedia.org/wiki/Basel_problem – user Dec 24 '17 at 21:50
  • Of course it is $;\pi^2/6;$ , @JohnWatson ...and I wouldn't dare to say there is no "elementary" proof in some sense. It is just that I know none – DonAntonio Dec 24 '17 at 21:50
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    I am amazed that many high-rep users here did not see one of the most famous topics on MSE! Merry Xmas to you all, by the way. – Jack D'Aurizio Dec 24 '17 at 21:51
  • Define "elementary", first. Curriculums of high schools vary. –  Dec 24 '17 at 21:52
  • Robert solution (on page 2) could be the most ''elementary'' approach. I'll have to think about it though. – nonuser Dec 24 '17 at 22:09
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    The Wikipedi article Basel problem tells how the problem was open for almost 100 years before Euler solved it. It Newton and Leibniz and other mathematicians could not solve it, what hope for an elementary solution? – Somos Dec 24 '17 at 23:31

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