I know that $\sum\limits_{n=1}^\infty n^{-2}=\pi^2/6$, but shouldn't the sum of rationals be rational? Is this akin to $\sum\limits_{n=1}^\infty n=-1/12$? Or does that mean that, somehow, $\pi^2/6$ is rational?
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Not necessarily an infinite sum.... – David G. Stork Mar 18 '20 at 06:00
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The sum of rationals, indeed, is rational. The limit of a sum of infinite rationals isn't necessarily, though – Saketh Malyala Mar 18 '20 at 06:00
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$3$ is rational. So is $3 + 0.1$. So is $3 + 0.1 + 0.04$. So is $3 + 0.1 + 0.04 + 0.001$. So is $3 + 0.1 + 0.04 + 0.001 + 0.0005$. But the limit of this series, $\pi$, is irrational.
Trevor Gunn
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You were both beaten by https://math.stackexchange.com/a/116270/42969 :) – Martin R Mar 18 '20 at 06:08
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The sum of rationals is rational, but $\sum_{k=1}^\infty \frac1{k^2}$ is not a sum of rational numbers. It's a series, and therefore the limit of a specific kind of sequencs: in this case, of the sequence $a_n=\sum_{k=1}^nk^{-2}$. Of course limits of sequences of rational numbers need not be rational.
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Here is an answer using the topology wording. Topology is the branch of mathematics that deals with continuity, limits, etc...
The value of a convergent series is the limit of its partial sums. As the set of rational numbers $\mathbb{Q}$ isn't closed, this limit can be in $\mathbb{R \backslash Q}$.
Besides, the second sum you mention $\sum n=-1/12$, is regularly seen on Math SE ; it is the equivalent of a mathematical (baseless) rumor...
Bibliography : If you want to know more about topology, see the downloadable book Topology without tears.
Jean Marie
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1Thank you very much for your answer and recommendation monsieur! And about the sum, yes, that is why I used "akin" in my wording; it intuitively seems absurd anyhow. – GDGDJKJ Mar 18 '20 at 06:18
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- not to say that mathematics is always in the range of the intuitive, but this one is a bit "too much".
– GDGDJKJ Mar 18 '20 at 06:21 -
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