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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.

  • 3
    Why don't you rewrite it as $f_n(x) = 1/(1/x^n + 1)$. Does that seem to help?2012-02-15
  • 2
    What have you tried so far? And do you want uniform convergence on the whole real line? Or perhaps on $[0,1]$ or $(1,\infty)$?2012-02-15
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    @belmont Any feedback on the answers??2012-02-27

2 Answers 2

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You can see that if we set

$${f_n}\left( x \right) = \frac{{{x^n}}}{{1 + {x^n}}}$$

$${f_n}\left( x \right) \to 1 \Leftrightarrow x>1 $$

$${f_n}\left( x \right) \to \frac{1}{2} \Leftrightarrow x=1 $$

$${f_n}\left( x \right) \to 0 \Leftrightarrow 0 \leq x < 1$$

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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.