We are aware of the limit $$ \lim_{n\to\infty}\sqrt[n]n = 1; $$ is there any geometric or otherwise intuitive reason to see why this limit holds?
Edit: I am adding some context, since this question was previously put on-hold, and I think one of the main reasons was that it was poorly motivated. From theorem 8.1 of Baby Rudin, suppose the series $$ \sum_{n=0}^\infty c_nx^n $$ converges for $|x|<R$, and define $$ f(x) = \sum_{n=0}^\infty c_nx^n \qquad (|x|<R). \tag{1} $$ Among other conclusions, the function $f$ is differentiable in $(-R,R)$, and $$ f'(x) = \sum_{n=0}^\infty nc_n x^{n-1} \qquad (|x|<R). \tag{2} $$ Rudin uses the fact that $\sqrt[n]n\to 1$ as $n\to\infty$ to justify that the series in $(1)$ and the series in $(2)$ have the same radius of convergence. I recognized the limit, but it is just such a nice combination of $n$ and the $n$th-root, that I thought there ought to be some nice intuitive way to understand it, hence this question.