What does $\sum_{n=0}^\infty 1/n^n$ converge to?

Question

Let $0^0 = 1$. It can be shown by the inequality introduced below that "my number" $\sum_{n=0}^∞ \frac{1}{n^n}$ is strictly between $2$ and $e$:

\begin{align} 2 & =\frac{1}{0^0}+\frac{1}{1^1} <\text{“my number''}=\frac{1}{0^0}+\frac{1}{1^1}+\frac{1}{2^2}+\frac{1}{3^3}+\cdots \\[10pt] & = \frac{1}{1}+\frac{1}{1}+\frac{1}{4}+\frac{1}{27}+\cdots<e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{6}+\cdots \end{align} Desmos.com (https://www.desmos.com/calculator/cj1bo9cvfx) shows that this number is equal to approximately 2.291. My questions are:

Can this number be expressed by any other way(s) than this one?
Is this number rational, algebraic irrational or transcendental?

$2.2912859970626635404072825905956\ldots$ is closer – Henry Jun 26 '17 at 23:37 — Henry, Jun 26 '17 at 23:37

score 3 · Answer 1 · answered Jun 26 '17 at 23:30

3

This sum is called the Sophomore's dream it can be expressed as an integral. It is not known if the value of the sum is rational, irrational or transcendental. $$\int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n} $$ Just add $+1$ if you want to include $n=0$.

answered Jun 26 '17 at 23:30

kingW3

13,496

1

by probability it is closer to be transcendental than any other kind of number :p – Masacroso Jun 26 '17 at 23:42
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It is irrational as any convergent series $\sum_{n=1}^\infty \frac{1}{\prod_{j=1}^n b_j}$ such that any prime $p$ divides some $b_j$ @Masacroso – reuns Jun 26 '17 at 23:47

What does $\sum_{n=0}^\infty 1/n^n$ converge to?

1 Answers1