2

$$ \sum_{p \, \in \text{ primes}} \frac{1}{2^p} $$

My friend came across a book which claimed this to be irrational and claimed it could be solved using the pigeonhole principle.

Bart Michels
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Agnishom Chattopadhyay
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    Well, the pigeonhole principle is used to prove that any rational number, when written in any base $b$, either terminates or repeats. This number, when written in base $2,$ does not terminate nor repeat. – Thomas Andrews Jan 05 '17 at 15:25
  • Dear Agnishom. You should first check the related questions (right margin), before posing a question. – Dietrich Burde Jan 05 '17 at 15:26
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    What do you think about the number $0.011010100010100010100010000010_2\cdots$ ? –  Jan 05 '17 at 15:26

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