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Uniform convergence of a sequence of functions.

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1. Show that the sequence x^2 (e^-nx) converges uniformly on [0, infinity).
2. Show that if a is greater than zero then the sequence (n^2 x^2 (e^-nx)) converges uniformly on the interval [a, infinity) but does not converge uniformly on the interval [0, infinity).

For problem 2 text gives a hint that if n is sufficiently large, ||f_n||_[a,infinity]=n^2 a^2/e^na , however ||f_n||_[0,infinity)=4/e^2 (A _ is used to denote a subscript).

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Solution Summary

Uniform cnfvergence of sequences of functions is examined and illustrated with two examples in the attached PDF file.

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Uniform Convergence of a Sequence of Functions

Prove the following theorem.

Let f1,f2,f3.... be continuous functions on a closed bounded interval [a,b] . Then fn--->f uniformly on [a,b] if and only if
fn(x)-->f(x) for every xn-->x such that xn,x E[a,b] .

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