I need to decide if the following function uniformly converges:
$$f_n(x) =\sqrt[n]{1+x^n} \quad, \quad x\in[0,\infty)$$ I found the sequence pointwise converges to
$$f(x) = \begin{cases} 1 \; , & \text{$x\in[0,1]$} \\ x \; ,& \text{$x \in(1,\infty)$} \end{cases}$$
I tried to find the supremum of $\;|f_n-f|\;$ but got nothing. Any ideas how I can get forward?
Hint. For $x>1$, $$ \sqrt[n]{1+x^n} - x = \frac{1}{x^{\frac{n-1}{n}}+x^{\frac{n-2}{n}}(1+x^n)^{\frac{1}{n}}+\cdots +x^{\frac{1}{n}}{(1+x^n)^{\frac{n-1}{n}}}} \le \frac{1}{n}$$ and for $x\le 1$, $$ \sqrt[n]{1+x^n} - 1 = \frac{x^n}{1+(1+x^n)^{\frac{1}{n}}+\cdots +{(1+x^n)^{\frac{n-1}{n}}}} \le \frac{1}{n}$$