convergence and limits How can I rewrite $\frac{1}{1+nx}$ and prove it's absolute convergence as $n \rightarrow \infty$? Given $\epsilon > 0$, should I define $f_n(x)$ and $f(x)$? 
Any help is hugely appreciated. Thank you

It seems you're being given $$f_n(x)=\frac{1}{1+nx}$$ and asked to find its (pointwise) limit as $n \to \infty$. Note that if $x>0$, $\lim f_n(x)=0$. If $x=0$, $f_n(x)=1$ for each $n$, so $\lim f_n(x)=1$. There is a little problem when $x<0$, namely, when $x=-\frac 1 n $, so I guess we just avoid $x<0$. Thus, you can see for $x\geq 0$, you function is $$f(x)=\begin{cases}1\text{ ; if} x=0\\0\text{ ; if } x>0\end{cases}$$
In particular, convergence is not uniform over $[0,\infty)$ (since the limit function is not continuous), nor in $(0,\infty)$ because $$\sup_{(0,\infty)}|f_n-0| =1$$
for any $n$. 
