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A constant without name?

  1. Aug 12, 2010 #1
    Hi,

    I was wondering if the constant
    [tex]
    \sum_{n=1}^\infty{2^{-2^n}}
    [/tex]

    has a certain name or some history or anything. It certainly appears not to have a closed form expression. It also certainly has some value because it's majorized by the simple geometric series. It's numerical value is 0.31642150902189314371 (given by http://www.research.att.com/~njas/sequences/A078585" [Broken], but is there anything else known about it?

    Regards,

    Pere
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 12, 2010 #2

    disregardthat

    User Avatar
    Science Advisor

    It's transcendental, if I'm not mistaken.
    [tex]|\sum_{n=1}^\infty{2^{-2^n}}- \sum_{n=1}^k{2^{-2^n}}| = |\sum_{n=k+1}^\infty{2^{-2^n}}| = |\sum_{n=1}^\infty{2^{-2^n2^k}}| \leq |\sum_{n=1}^\infty{2^{-2^n}}|^{2^k} < \left(\frac{1}{2}\right)^{2^k}=\frac{1}{2^{2^{k+1}}}[/tex]

    The denominator of the rational number [tex]\sum_{n=1}^k{2^{-2^n}}[/tex] is [tex]2^{2^k}[/tex]. The number is thus a liouville number, and therefore transcendental.
     
    Last edited by a moderator: May 4, 2017
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