6

I wish to know the exact value of $$\sum_{n=1}^\infty\frac1{n^n}.$$

I've found on the internet some mentions of the equality (the Sophomore's Dream, as I've learned) $$\sum_{n=1}^\infty\frac1{n^n} = \int_0^1\frac1{x^x}dx.$$

Wolfram Alpha's computations returns the value of $1.29128599706266\dots$ to the RHS integral. However, it says no result found in terms of standard mathematical functions for $$\int\frac1{x^x}dx$$ so the equality does not yield a definitive answer.

I am aware there might be no better way to describe this constant value, and it could just be given its own name, such as the Euler–Mascheroni constant. Wondering if that was the case I've searched for its first few digits and was led to the Sophomore's Dream and then to this and this related questions, none of which relieved my curiosity. All I've learned was that it was not even known if this constant is a rational number.

I apologize for the nature of my question, which is utterly meta, but I just cannot help myself to wonder: Is this problem uninteresting? Why were so many mathematicians mobilized to solve the Basel Problem but so little is talked about this (surely just as pretty) series? Is this somehow a much harder question? Why?

Also, I'm guessing if this value had a name someone would have mentioned it by now, so I'm calling it Bernoulli's Constant (not Sophomore's Constant since there are two equations in the Sophomore's Dream).

The digit sequence is A073009 on OEIS, as pointed out in a comment.

Alma Arjuna
  • 3,759
  • 3
    Perhaps someone is working on finding the value for it, and plans to put it in a publication, but they don't want to tell others at risk of getting scooped. – WDR Feb 25 '24 at 19:12
  • 1
    @WDR but the Sophomore's Dream exists since $1697$ – Alma Arjuna Feb 25 '24 at 19:14
  • 1
    Fermat's last theorem went unproven for hundreds of years before it was solved, perhaps this problem is really difficult. – WDR Feb 25 '24 at 19:15
  • 16
    Mathematicians have played around with it. Nothing could be figured out. The number is almost certainly transcendental given how fast the denominator grows (I am surprised to hear it's not known to be irrational because of that). Most transcendental numbers lack a closed form expression, so there's no reason to hope your expression would have one. It would be considered quite miraculous if a closed form in terms of simpler functions could be found, but there's little reason to spend time on a very likely impossible problem when there are plenty of problems in math to be solved. – Brevan Ellefsen Feb 25 '24 at 19:29
  • 4
    In regards to you comparison to the Basel problem, it has connections to number theory and the Riemann Zeta functions, which are interesting in their own right. This has no such interesting connections to my knowledge – whpowell96 Feb 25 '24 at 19:33
  • 2
    How about https://oeis.org/A073009? – m-stgt Feb 25 '24 at 19:38
  • 3
    "Is this problem uninteresting?" Honestly? Yes. By far nothing like Basel problem. You've linked wiki article yourself, it explains quite well why it is interesting. – freakish Feb 25 '24 at 19:42
  • 1
    @freakish I don't get it, why would that be? – Alma Arjuna Feb 25 '24 at 19:48
  • 1
    My guess is it is because it is widely conjectured that there is no "closed formula" for this number. – Peter Feb 25 '24 at 19:55
  • 2
    You can “invent” the sophomore dream function – Тyma Gaidash Feb 25 '24 at 20:08
  • 4
    @AlmaArjuna Basel problem serves other parts of maths, e.g. calculating probabilities of random numbers being coprime. So it has applications. What applications has this series? It may be interesting to you, but for mathematicians to spend time on it, there has to be better reason than just stylish symbols. – freakish Feb 25 '24 at 20:11
  • 1
    Perhaps "people are trying to find the value" ... but they usually do not report their failures. – GEdgar Feb 26 '24 at 06:03

0 Answers0