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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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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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    @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.2011-10-31