What is the name of this function $f(x) = \frac{1}{1+x^n}$? $f(x\in\mathbb{R}) = \frac{1}{1+x^n}$ 

In the particular case where $n$ is even, this looks like the pdf of the Cauchy distribution, so you might want to say that $f(x) = \frac{1}{1+x^{2n}}$ is some kind of a generalized Cauchy...
