Testing if $f_n(x) = \sqrt[n]{x}$ converges uniformly.

Question

Given: $$f_n(x) = \sqrt[n]{x} \; , x \in [0,1]$$

Is the above series of functions converge uniformly? and how do we check it?

Answer 1

One more way of seeing it:

Take $x \in (0,1]$. Then $\lim_{n \to \infty} x^{\frac{1}{n}}=1.$ Now take $x=0$. $\lim_{n \to \infty}x^{\frac{1}{n}}=0 \ne 1$. Hence the function does not converge uniformly, but it converges uniformly for $x \in (0,1]$

Answer 2

Hints:

$$\sqrt[n] a\xrightarrow[n\to\infty]{}1\;,\;\;\forall\,\,0<a\in\Bbb R$$

$$\sqrt[n] 0=0$$