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Uniform convergence

Given the sequence of functions where such that

Prove that

(i) the serieses and converge uniformly to functions .

(ii) the functions are continuous.

(iii) .

(iv) it holds that

Solution

(i) This is an immediate consequence of the Weierstrass M Test . We simply note that and of course

(ii) The uniform limit of continuous functions is continuous. This is enough for us to extract that both and throughout .

(iii) For a limit of differentiable functions, a sufficient condition for claiming that the derivative of the limit is the limit of the derivatives is the uniform convergence of the sequence of derivatives. Since the series for converges uniformly, we can claim that .

(iv) For the first integral we have that

and for the second integral we have

since the function is odd.