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If $p_i$ is an infinite set of distinct primes such that $c=\sum\frac{1}{p_i} < \infty$, must $c$ be a transcendental number?

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up vote 13 down vote accepted

No. Pick any positive algebraic number, and by choosing your primes carefully, you can make the infinite series converge to your number.

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Is it obvious that this can be done ? Could you give a reference to a book/article discussing this ? Thanks! – Sasha Oct 30 '11 at 21:45
The sum $\sum \frac{1}{p_i}$ diverges. So for any $k$, the sums $\sum_{k}^{k+n}\frac{1}{p_i}$ are unbounded. Also, the $\frac{1}{p_i}$ approach $0$. That's all we need. Pick any positive real $x$. Add up enough $\frac{1}{p_i}$ so that we get within $1$ of $x$, but below it. Then add enough more $\frac{1}{p_i}$ so we get within $1/2$ of $x$, but below it. And so on. – André Nicolas Oct 30 '11 at 21:58
@Andre thanks. Indeed, this proves that if $a_i$ are positive and approaching zero with divergent sum then there's a subseries converging to any given positive real. – Gerry Myerson Oct 31 '11 at 0:39

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