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What is the summation of this infinite series $$1 + \frac{1}{3} + \frac{1}{7} + \frac{1}{15} + \frac{1}{31} + \frac{1}{63} + ... ?$$

zoli
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Sukhyun Park
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    A starting point is to find the lower bound and upper bound. What does the limiting value of the infinite sum $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}$ tell you? – Toby Mak May 20 '17 at 01:05
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    It has no closed form without using 'special' functions like digamma, but it's a well-known constant: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant – Steven Stadnicki May 20 '17 at 01:06
  • The sum of the receiprocals of mersenne numbers. That's what I am looking for. But is there an exact value other than an approximate value? – Sukhyun Park May 20 '17 at 01:20
  • On @StevenStadnicki 's Wikipedia page it mentions that Erdos proved that the number is irrational. The OEIS page (https://oeis.org/A065442) does not mention if it is transcendental. – Toby Mak May 20 '17 at 01:26
  • I wouldn't be surprised if transcendentality is provable by showing that it's too well-approximable. – Steven Stadnicki May 20 '17 at 01:44
  • @StevenStadnicki: here (https://math.stackexchange.com/a/1978329/44121) I showed a series acceleration technique that should easily lead to the trascendentality of such constant. – Jack D'Aurizio May 20 '17 at 12:39

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