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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-:https://en.wikipedia.org/wiki/Basel_problem. How can the sum of rationals be equal to an irrational number?

Arihant Jain
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    Induction doesn't apply to infinite sums. You know by induction that any finite sum of rationals is a rational. – Thomas Andrews Feb 10 '16 at 17:39
  • Sum of finitely many rationals is rational. – Apurv Feb 10 '16 at 17:40
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    See http://math.stackexchange.com/questions/116269/how-can-adding-an-infinite-number-of-rationals-yield-an-irrational-number. – Eli Rose Feb 10 '16 at 17:41
  • Note that every irrational number is an infinite sum of rational numbers (let $x$ be irrational and let's assume that $0 < x < 1$ let $x_k$ be the $k-th$ decimal place of $x$. Then $x = \sum_{k=1}^\infty x_k\cdot 10^{-k}$ and each $x_k \cdot 10^{-k}$ is rational. – Stefan Mesken Feb 10 '16 at 17:43
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    An easy example of rationals adding up to an irrational:$$\pi=3+0.1+0.04+0.001+0.0005+\dotsb$$ – Akiva Weinberger Feb 10 '16 at 17:51
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    The key thing to note is that an infinite sum is not just "a sum." It is a limit of a bunch of sums. Lots of things you expect from finite sums don't work in infinite sums - re-arrange an infinite sum, for example, and you get different results. – Thomas Andrews Feb 10 '16 at 18:12
  • A series is not a sum. – Hagen von Eitzen Feb 10 '16 at 18:12

2 Answers2

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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.

Jack M
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