I'm having difficulty determining where this sequence of functions $\displaystyle f(n,x)=\frac{x^n}{(1+x^n)}$ converges, and whether it converges uniformly.
Thanks.
I'm having difficulty determining where this sequence of functions $\displaystyle f(n,x)=\frac{x^n}{(1+x^n)}$ converges, and whether it converges uniformly.
Thanks.
You can see that if we set
${f_n}\left( x \right) = \frac{{{x^n}}}{{1 + {x^n}}}$
${f_n}\left( x \right) \to 1 \Leftrightarrow x>1 $
${f_n}\left( x \right) \to \frac{1}{2} \Leftrightarrow x=1 $
${f_n}\left( x \right) \to 0 \Leftrightarrow 0 \leq x < 1$
not on the entire $\mathbb{R}$ of course, since the limit function is not continuous.
and it is uniform on $[0, 1-\varepsilon]$ and on $[1+\varepsilon, +\infty)$ by compaire it with $(1-\varepsilon)^n$ and $(1+\varepsilon)^n/(1+(1+\varepsilon)^n)$ resp.