5
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As far as I understand most of these questions use the M-test, but I can't find a series that suffices.

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Wait. $\lim\limits_{x\to0}f_n(x)=0$, but $\lim\limits_{x\to0}f_\infty(x)=1$. There is no uniform convergence!

Apparently, $f_\infty(x)=\sqrt{x+{1\over4}}+{1\over2}$

You might want to show the uniform convergence on $(1,\infty)$ or $(\varepsilon,\infty)$; that's another story, and a simple one at that.

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  • $\begingroup$ Interesting. maybe the question is flawed. How did you gather that $\lim\limits_{x\to0}f_n(x)=0$ ? $\endgroup$ – PanthersFan92 Jun 28 '16 at 12:53
  • $\begingroup$ @PanthersFan92 you can do this by induction on $n$, $f_n(0)=0$ and $f_{\infty} (0) =1$ $\endgroup$ – clark Jun 28 '16 at 12:57

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