Analysis II | Part IB, 2003

Explain what it means for a sequence of functions (fn)\left(f_{n}\right) to converge uniformly to a function ff on an interval. If (fn)\left(f_{n}\right) is a sequence of continuous functions converging uniformly to ff on a finite interval [a,b][a, b], show that

abfn(x)dxabf(x)dx as n\int_{a}^{b} f_{n}(x) d x \longrightarrow \int_{a}^{b} f(x) d x \quad \text { as } n \rightarrow \infty

Let fn(x)=xexp(x/n)/n2,x0f_{n}(x)=x \exp (-x / n) / n^{2}, x \geqslant 0. Does fn0f_{n} \rightarrow 0 uniformly on [0,)?[0, \infty) ? Does 0fn(x)dx0\int_{0}^{\infty} f_{n}(x) d x \rightarrow 0 ? Justify your answers.

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