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

Not every function deserves a name beyond the one it defines for itself by merely describing it (like in Ethan's answer).
EDIT: I would emphasise the spacing to avoid ambiguity:

One over $\quad$ one plus $\quad x$ to the $n$th power.

