Show the sequence converges uniformly?

Question

I was trying out some problems in the textbook and I came across this one which is bit tricky for me. Would anyone please be kind enough to help me out?

Show that the sequence $\sum_{n=1}^\infty \frac{nx}{5n-x+1}$ converges uniformly to $f$ on [0,3] where $f(x)=\frac{x}{5}$

Thank you

Answer 1

$f_n(x)=\frac{nx}{5n-x+1}=\frac{x}{5-\frac{x}{n}+\frac{1}{n}}\rightarrow \frac{x}{5}$ for n$\rightarrow \infty$

So $f(x)=\frac{x}{5}$

Now $M_n=sup\{|f_n(x)-f(x)|:x\in [0,3]\}$

$|f_n(x)-f(x)|=\frac{x^2-x}{5(5n-x+1)}$ which will definitely give a maximum value for some $x\in [0,3]$ But in the denominator we have $n$ So $M_n\rightarrow 0$ as $n\rightarrow \infty$ So sequence converges uniformly