0
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ Not necessarily an infinite sum.... $\endgroup$ Commented Mar 18, 2020 at 6:00
  • $\begingroup$ The sum of rationals, indeed, is rational. The limit of a sum of infinite rationals isn't necessarily, though $\endgroup$ Commented Mar 18, 2020 at 6:00

3 Answers 3

Reset to default
4
$\begingroup$

$3$ 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.

$\endgroup$
2
  • $\begingroup$ +1 Beat me to it. I'll delete mine. $\endgroup$ Commented Mar 18, 2020 at 6:05
  • $\begingroup$ You were both beaten by math.stackexchange.com/a/116270/42969 :) $\endgroup$ Commented Mar 18, 2020 at 6:08
1
$\begingroup$

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.

$\endgroup$
5
  • 1
    $\begingroup$ Thank 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. $\endgroup$ Commented Mar 18, 2020 at 6:18
  • $\begingroup$ * not to say that mathematics is always in the range of the intuitive, but this one is a bit "too much". $\endgroup$ Commented Mar 18, 2020 at 6:21
  • 1
    $\begingroup$ I agree. Our mind isn't well "programmed" to deal with infinity :) $\endgroup$ Commented Mar 18, 2020 at 6:21
  • 1
    $\begingroup$ About counterintuitive results in mathematics, see for example $\endgroup$ Commented Mar 18, 2020 at 6:25
  • 1
    $\begingroup$ See also this one or, this one in another "range" $\endgroup$ Commented Mar 18, 2020 at 6:48
1
$\begingroup$

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.

$\endgroup$
2

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.