Show that the series for which the sum of first $n$ terms $f_n(x) = \frac{nx}{1+n^2x^2}$ cannot be differentiated term-by-term at $x = 0$. What happens at $x \neq 0$?
As-
Here we are given with the sequence of partial sums. We know that $(f_n) \to f $ uniformly on $S$ iff $$ \lim [ \sup\{ |f(x) - f_n(x) : x \in S \} ] = 0 \; \; $$Here note that $f \equiv 0$. $$ |f_n - f | = f_n = \frac{nx}{1 + n^2x^2 } \implies f_n'(x) = \frac{n(1 + n^2x^2) - nx(2xn^2)}{(1 + n^2x^2)^2}$$ and $$ f'_n(x) = 0 \iff x = \frac{1}{n}, \frac{-1}{n} $$ Now observe that $$ f_n\bigg( \frac{1}{n} \bigg) = \frac{1}{2} \implies \lim \sup|f_n - f| = \frac{1}{2} \neq 0$$Therefore we conclude that $f_n$ does not converges uniformly to $0$ and hence we can say that series can't be differentiated term by term.
Am i correct? What about any other point?