Let $\sum_{n=0}^{∞}\frac{1}{x^{n!} } = 1$ . Is x a known number at all? Can it be proven to be rational/irrational? 

I was thinking about some infinite sums when I thought about this infinite, recursive sum in the form of an equation. I'm really intrested in knowing more about a number like x, so feel free to tell me everything you may know about it. Hope someone feels as curious about this number as me. Good luck :)

2 Answers

Interesting question!

Mathe Mathe
3.2K
  • This number does exist! The sum does not start at one, it starts at 1/x since 0!=1. It is possitive and it's ~2.44, but that's pretty much all I can decipher about this weird, seemingly irrational number.

    • Mathe Mathe
      +1

      You are right, the first number is 1/x !

One can easily find an approximation, x ≈ 2.4244104490156532363723745970794970842, for example with this simple PARI code :

 solve(x=2,3,sum(n=0,19,x^(-n!))-1)

which you can paste into the online PARI/gp interpreter.
Yes, it's sufficient to take the sum up to 19, or even only 9, instead of ∞, to get the same 40 digits, since 9! ~ 10⁸ and  2.4 ^ 9! is already *much* more than 10^40.
I'm certain that the number is irrational, but I don't have any idea how one could prove this.

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