Which of the following sequences/series of functions are uniformly convergent on $[0,1]$.

Question

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 continuous.so it is not uniformly continuous. (b) true by $M$ test. (c) no idea.

can anybody help me please

Answer 1

1. limit function is discontinous hence not UC.

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

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