Is there any function that has the series expansion $x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+\cdots$? $$\frac{1}{1-x} = 1+x+x^2+x^3+ \cdots$$
Is there a $f(x)$ that has the series of $n$th roots?
$$f(x)= x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+ \cdots$$
Wolfram Alpha seemed to not understand my input.

For $x>0,~x+x^{\frac{1}{2}}+x^{\frac{1}{3}}+...$ can't converges since $x^{\frac{1}{n}}\to1.$ Neither does it for $x<0$ since $x^{\frac{1}{2}}$ is undefined then. So the only possibility is the trivial one: $$f:\{0\}\to\mathbb R:0\mapsto 0$$
