What number/function/thing to the power n gives n itself

Question

Is there a number/function/thing if put to the power n gives n? I know that there is log to the base 2 of 2n gives n. which is log2(2n)=n But i am asking for (thing)n=n. In fact i want an x=thing that if plugged in the formula: 1+x2 +x3 +x4 ........ =1/(x-1) gives me 1+2+3+4+5.....=1/(thing-1) I would be grateful if you could help me

Answer 1

If you assume $n$ is a positive integer, and the "thing" you're looking for is a real number, then you want to solve $x^n = n$. But the only solution (up to a sign change, for even $n$) is $\sqrt[n]{n}$, which won't work for all $n$, only a particular one.

However, you might find this interesting: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF