For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number.
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But for example $$\pi=3+0.1+0.04+0.001+0.0005+0.00009+0.000002+\cdots$$ and that surely does not seem strange to you... |
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You can't add an infinite number of rational numbers. What you can do, though, is find a limit of a sequence of partial sums. So, $\pi^2/6$ is the limit to infinity of the sequence $1, 1 + 1/4, 1 + 1/4 + 1/9, 1 + 1/4 + 1/9 + 1/16, \ldots $. Writing it so that it looks like a sum is really just a shorthand. In other words, $\sum^\infty_{i=1} \cdots$ is actually kind of an abbreviation for $\lim_{n\to\infty} \sum^n_{i=1} \cdots$. |
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Others have demonstrated some examples that make clear why this can happen, but I wanted to point out the key mathematical concept here is "Completeness" of the metric space. A metric space is any set with "distance" defined between any two elements (in the case of $\mathbb{Q}$, we would say $d(x,y) = |x-y|$). A sequence $x_i$ is "Cauchy" if late elements stop moving around very much, a necessary condition for a sequence to have a finite limit. To put it formally, ${x_i}$ is cauchy for $\epsilon>0$, there is a sufficiently large $N$ so that for every $m,n>N$ we have $d(x_n,x_m)<\epsilon$. A metric space is complete if all Cauchy sequences have a limit in the space. The canonical complete metric space is $\mathbb R$, which is in fact the completion of $\mathbb{Q}$, or the smallest complete set containing $\mathbb Q$. We think of an infinite sum as the limit of a sequence of partial sums: $$\sum_{n=1}^\infty x_n = \lim_{N\to\infty}\left( \sum_{i=1}^Nx_n \right)$$ As others have pointed out with a number of good counter-examples (my favorite of which is the decimal representation of an irrational number), $\mathbb{Q}$ is not complete, therefore an infinite sum of elements of $\mathbb Q$, for which partial sums are necessarily elements of $\mathbb Q$, can converge to a value not in $\mathbb Q$. |
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It is counter-intuitive only if you are adding a "finite" number of rational numbers. Otherwise, as @Mariano implied, any irrational number consists of an infinite number of digits, and thus can be represented as a sum of rational numbers. |
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Besides that. Given a sequence of positive rational numbers such that their sum converges and such that $a_n > \sum_{k = n + 1}^\infty a_k$. Then choosing a $\pm$ sign for each term of the sequence gives a new convergent series, each to a different number. By a countable-uncountable argument you get nonnumerable examples of that kind of series :). |
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This has to see with the rate at which the sum/series converges to its limit and the Roth-Thue-Siegel theorem which allows you to use the rate of convergence to decide if the limit is rational or not. Maybe Emile (the OP) meant to ask something of this sort (please let me know, Emile): why do some (convergent, of course) infinite sums of rationals are rational and others are irrational? |
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