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$\cfrac{1}{1^2} + \cfrac{1}{2^2} + \cfrac{1}{3^2} + ...... \infty = \cfrac{{\pi}^2}{6}$

Often termed as one of the most beautiful equations in math:

I tried devising a proof for the above formula.

But I got stuck just as soon as I begun. Addition of any two or more rational numbers always results in a rational number. The $LHS$ is rational but the $RHS$ is irrational. How is this possible?

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    A convergent sequence of rational numbers can have an irrational limit. That's how this equality is possible. – avs Feb 03 '21 at 06:08
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    Adding finitely many rational numbers yields a rational, but with infinitely many anything is possible. – N. S. Feb 03 '21 at 06:08
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    https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro – Riquelme Feb 03 '21 at 06:09
  • Evaluating the sum on the left is called the Basel problem. Euler's beautiful, non-rigorous solution was one of the highlights of the history of math. https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro – littleO Feb 03 '21 at 06:12
  • Thanks for the reference. –  Feb 03 '21 at 06:12
  • The Youtube channel 3Blue1Brown (which any Math enthusiast probably knows) has a beautiful intuitive explanation on this result. Perhaps complimenting the technical proofs suggested by others with such an intuition is a good idea : ) – JohannesPauling Feb 03 '21 at 06:24
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    Why do people put $\infty$ there as a summand? – user10354138 Feb 03 '21 at 06:42

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At least as far as your question is concerned, take your favorite irrational number. For instance we can take $$\pi(=3.1415...)$$ and write it as an infinite convergent sum $$ \pi=3+\frac{1}{10}+\frac{4}{100}+\frac{1}{1000}+\frac{5}{10000}+\cdots $$ So actually, every irrational can be viewed as the sum of an infinite series consisting of rational numbers.

Alekos Robotis
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