How can I show that this sequence of derivatives converges uniformly on [-R,R] for $R>0$

Question

I am trying to show that the sequence of derivatives converges uniformly on [-R,R] for and $R>0$. The problem is that the sequence of functions is: $$f_{n}(x)= \frac{nx + x^2}{2n} $$ and when I take the derivative to get $f'_n(x)$ I get the following: $$ f'_n(x) = \frac{n + 2x}{2n} $$ which will then simplify to: $$f'_n(x) = 1/2 + \frac{x}{n}$$ but when I do the ratio test or the root test for these I get a value of 1 but I also get that $$\lim _{n\rightarrow \infty }\left| { 1 + \frac{x}{n} } \right| = \infty $$ any help would be greatly appreciated.
Note I did try to use the M-test but unless I am picking an incorrect $M_n$ then I couldn't get it to work as well.

Answer 1

It is clear that $(f_n')$ converges pointwise t0 $1/2$. Furthermore:

$|f_n'(x)- \frac{1}{2}|=\frac{|x|}{n} \le \frac{R}{n}$ for $x \in [-R,R]$.

Your turn !