In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$
here $\sum u_n'(x)$ is not uniformly convergent, BUT
If $f '(x) = \lim_{n\to\infty} f_n'(x) $
where $f(x)=\lim_{n\to\infty} f_n(x) $
Then this series can be differentiated term by term, is this condition true everywhere for any series?
But the same condition fails for a series whose $ f_n(x)=\dfrac{nx}{1+n^2x^2} $, this cannot be differntiated term by term at $x=0$, WHY?
Can we do term-by-term differentiation of a series which is not uniformly convergent?? If so what is the condition to be satisfied there?