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Which of the following sequences/series of functions are uniformly convergent on $[0,1]$.
(a) $f_n(x)=(cos(\pi n!x))^{2n}$.
(b) $∑_{n=1}^∞ \cos (m^6x)/m^3$.
(c) $f_n(x)= n^2x(1-x^2)^n$

(a) the limit function is not it is not uniformly continuous. (b) true by $M$ test. (c) no idea.

can anybody help me please

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(c) $f_n(1/\sqrt{n})\to\ell$ with $\ell=$ $____$? – Did Jan 20 '13 at 11:00
  1. limit function is discontinous hence not UC.

  2. UC as $\le\frac{1}{m^3}$ and apply M-test.

  3. not UC, as $\lim_{n\rightarrow\infty}\int_{0}^{1}f_n(x)dx\neq\int_{0}^{1}f(x)dx$

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