The problem is:
$f_n(x)=\sum_{j=1}^{n} \frac{x}{1+j^2x^2} (n=1,2,…,x>0)$,$f(x)=\lim \limits_{n\rightarrow \infty}f_n(x)$
$1$.Show that $f(x)$ exist and prove $\frac{\pi}{2}-\arctan x\leq f(x)\leq \frac{\pi}{2}-\arctan x+\frac{x}{1+{x}^{2}}$
$2.$Determine whether $f_n(x)$ uniformly converge to $f(x)$ or not and give a proof.
For the first, $$\bigg|\frac{x}{1+j^2x^2}\bigg| \leq \frac{1}{x} \frac{1}{j^2},\frac{1}{x} \sum_{j=1}^{n}\frac{1}{j^2} \leq 2\frac{1}{x}$$and $f_n(x)$ increases with n, so $f$ exists. As $$\arctan x=\sum_{j=1}^{n} \frac{\frac{x}{n}}{1+{(\frac{j}{n})}^2x^2},\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}$$
I tried with $$\omega=\frac{1}{x}, \sum_{j=1}^{n} \frac{x}{1+j^2x^2}=\sum_{j=1}^{n} \frac{\omega}{\omega^2+j^2}$$ but cannot get the relation between $\frac{\frac{x}{n}}{1+{(\frac{j}{n})}^2x^2}$ and $\frac{x}{x^2+j^2}$.And I do not find condition for useing Abel or Dirichlet for the second question. How should I deal with these two question? Any valuable suggestions are welcome.
About uniform convergence, you don't have any hopes. If the sequence uniformly converged, you should have that $||f_n-f_m||$ is as small as you like as $n,m$ are big enough.
Instead, try to show that for every $N$ there exist $n,m > N $ ("big enough") such that $f_n - f_m> 1 $ at some point $x_0$.