How can I use Weierstrass test to determine a set $I$ on which the series $\displaystyle \sum_{n=1}^{\infty} nx^n(1-x)^n$ converges uniformly?
Weierstrass test: Let $f_n : I \to \mathbb{R}$ be a sequence of functions with $|f_n(x)| \le M_n$ for all $x \in I$ and $k=1,2, \dots$. If $\sum_n M_n$ converges then $\sum_n f_n$ converges uniformly on $I$.