I learned "one way only" regarding how to prove uniform convergence of a series. This way is using normal convergence, and if the series is not normally convergence, then check $\|f_n\|_\infty$, if it's tending to zero then check $\|R_n\|_\infty$ ( if $R_n$ converges uniformly to $0$) , if it's tending to zero then the series is uniformly convergent.
Ok, I tried this here. I proved the pointwise convergence. Then I proved that there is no normal convergence since $\sum_{k=0}^\infty \|f_k\|_\infty$= $\sum_{k=0}^\infty \frac{1}{2k}$ which's divergent.
And since $\|f_n\|_\infty$ = $\frac{1}{2n}$ $\longrightarrow$ $0$, we still have no result so we should check $\|R_n\|_\infty$:
$|R_n(x)|$= $|\sum_{k= n+1}^\infty f_k(x)|$ $\leq$ $\sum_{k= n+1}^\infty |f_k(x)|$ $\leq$ $\sum_{k= n+1}^\infty \frac{x}{n^2}$ $\leq$ $c$ where $c$ is a constant.
Checking if $R_n$ is a decreasing sequence: $R_{n+1}(x) - R_n(x)$ = $-f_{n+1}(x)$ $<$ $0$ if $x>0$. So $R_n$ is a decreasing and bounded sequence which gives us that it tends to $0$. But in the link I put, they proved the non-uniform convergence. So can anybody tell me what is the wrong part of my solution with explanation please :)