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I know there are many ways to prove that the sum of the reciprocals of the primes diverges, but does the following argument work?


The cardinality of the set of all prime numbers is obviously ${\aleph_0}$. Intuitively we can map each natural number $N$ to a prime number ${p_N}$. Therefore $$\sum\limits_{n = 1}^\infty{\frac{1}{{p_n}}}$$ diverges because \begin{align} 1 &\mapsto {p_1}\\ 2 &\mapsto {p_2}\\ 3 &\mapsto {p_3}\\ &\vdots\\ \end{align} i.e. it resembles the harmonic series $$\sum\limits_{n = 1}^\infty{\frac{1}{n}}.$$ If it does not work, is there a way to make this argument work?


Edit: If this argument does not work in general, why does it make intuitive sense for the primes, but not for the reciprocals of the squares?

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    Notice that for every $i,$ we have $\dfrac{1}{p_i}<\dfrac{1}{i},$ which doesn't help for the comparison test.2012-12-14
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    The convergence of harmonic subseries is related to the density of its terms. See the work of Šalát et al; for instance http://eudml.org/doc/118201.2012-12-14
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    The question, "why does it make intuitive sense for the primes," is a question of psychology (specifically, a question of *your* psychology), not of mathematics.2012-12-14
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    @GerryMyerson Almost all theorems are motivated by intuition and experience. We are not machines.2012-12-14
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    True. But if that was meant as a reply to my comment, I don't see the connection.2012-12-14

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No, it doesn’t. If the argument worked, it would prove that $\sum_{n\ge 1}\frac1{n^2}$ diverges, since you could set up a similar correspondence.

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    Is there a way to make this argument work?2012-12-14
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    @glebovg: Not that I can see: cardinality is just too crude a measure to be very useful here.2012-12-14
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    Brian, do you think you can add something regarding my edit?2012-12-14
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    It does not "make intuitive sense." It is not that the argument does not work "in general." It just does not work. It so happens that some series diverge (such as $\sum\frac1{p_i}$), but the reason is not that they have infinitely many terms.2012-12-14
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    @glebovg: I agree with Andres: it doesn’t make intuitive sense to me. I can often see the thinking behind students’ mistakes, but here I have to admit that can’t see why you feel that it does make intuitive sense.2012-12-14
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    So you do not intuitively think that the sum of the reciprocals of the primes diverges, and the sum of the reciprocals of the squares does not?2012-12-14
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    @glebovg: I was talking about the argument that you offered, not about the fact itself. But no, I don’t think that it’s intuitively obvious that the sum of the reciprocals of the primes diverges.2012-12-14
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    If you want something fun to think about, consider the sum of the reciprocals of those positive integers with no $9$ in their decimal representations. What does your intuition tell you about them?2012-12-14
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    @Gerry: Oh, good one!2012-12-14
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    @GerryMyerson Do you mean Kempner series? Yes, intuitively it should diverge but it does not.2012-12-14
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    I didn't recall that Kempner was associated to that series.2012-12-14
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    @Gerry: He seems to have given the first published proof of its convergence; see [the Wikipedia article](http://en.wikipedia.org/wiki/Kempner_series). MathWorld uses the name, for whatever that’s worth.2012-12-14