Test the uniform convergency of a series of function Consider the series of function $$\sum_{n=1}^{\infty}\frac{x}{1+n^2x}.$$ Show that this series of function is NOT uniformly convergent in $[0,1]$. I know only two methods to show a series of function to NOT uniformly convergent : (i) Using definition of uniform convergent. (ii) Showing that term by term integration is not possible. i.e. showing that$$\int_0^1\sum_{n=1}^{\infty}f_n(x)\,dx\not =\sum_{n=1}^{\infty}\int_0^1f_n(x)\,dx.$$ But in each case we need to find out the finite sum $\sum_{k=1}^{n}f_k(x)$. I am not able to find out this finite sum. Please help on it.... Does there any other process or theorem to find that the series is not uniformly convergent ?

You may observe that, with $\displaystyle f_n(x)=\frac{x}{1+n^2x}$, $x \in [0,1]$, we have $$ (f_n(x))'=\frac{1}{(1+n^2x)^2}>0 $$ thus $\displaystyle f_n(x)$ is increasing with $x$: $$ 0<\sup_{[0,1]}f_n(x)=f_n(1)=\frac{1}{1+n^2} \leq \frac{1}{n^2}, $$since $\displaystyle \sum\frac{1}{n^2}$ is convergent, hence **your initial series is uniformly convergent in $[0,1]$.**