Define what it means for a sequence of functions $F_{n}:(0,1) \rightarrow \mathbb{R}$, where $n=1,2, \ldots$, to converge uniformly to a function $F$.

For each of the following sequences of functions on $(0,1)$, find the pointwise limit function. Which of these sequences converge uniformly? Justify your answers.

(i) $F_{n}(x)=\frac{1}{n} e^{x}$

(ii) $F_{n}(x)=e^{-n x^{2}}$

(iii) $F_{n}(x)=\sum_{i=0}^{n} x^{i}$