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I have seen proof of -: Sum of two rationals is rational and that implies that sum of all rational is always rational( by induction). Now my question is about Basel Problem-:. How can the sum of rationals be equal to an irrational number?

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2 Answers

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You're interpreting the sum wrong. Actually, the sum means

$$\sum_{k=1}^\infty \frac 1{k^2}=\lim_{N\to\infty}\sum_{k=1}^N \frac 1{k^2}$$

And the limit of a sequence of rational numbers can definitely be irrational.

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The sum of two rationals is rational, therefore, by induction the sum of a finite number of rationals is rational, but induction can't prove anything about sums of infinitely many rational numbers.

Induction tells you that if something is true for $1$, and if it being true for $n$ implies it's true for $n+1$, then it's true for all finite $n$. It doesn't allow you to say anything about infinity. For instance, you can prove by induction that the sum of $n$ numbers is finite. That doesn't mean that the sum of infinitely many numbers is always finite.

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