A constant without name?

  • #1

Main Question or Discussion Point

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:

Answers and Replies

  • #2
disregardthat
Science Advisor
1,854
33
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
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:

Related Threads for: A constant without name?

Replies
2
Views
1K
Replies
7
Views
1K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
5
Views
1K
Top