Analysis II | Part IB, 2004

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

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

(i) Fn(x)=1nexF_{n}(x)=\frac{1}{n} e^{x}

(ii) Fn(x)=enx2F_{n}(x)=e^{-n x^{2}}

(iii) Fn(x)=i=0nxiF_{n}(x)=\sum_{i=0}^{n} x^{i}

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