This is just my comment reposted as an answer.

The problem of which functions $f:[0,1]\to\mathbb{R}$ can be approximated by polynomials with non negative coefficients has already been studied, eg by Robert Whitley: as theorem 2 of [this paper][1] states, a continuous function $f:[0,1]\to\mathbb{R}$
 is a pointwise limit of polynomials with non negative coefficients iff $f(x)=\sum_{n=0}^\infty a_nx^n$ for some non negative sequence $a_n$ such that $\sum a_n<\infty$.


  [1]: https://www.sciencedirect.com/science/article/pii/0021904576901209