4. The minimum value of the sequence $\left\{\frac{1}{\sqrt[n]{n}}\right\}, n=1,2, \cdots$, is
$y=\frac{1}{\sqrt[n]{n}} \Rightarrow \ln y=-\frac{1}{n} \ln n$, Let $f(x)=-\frac{\ln x}{x}, f^{\prime}(x)=-\frac{1-\ln x}{x^{2}} \Rightarrow f(x)$ is monotonically decreasing on $(1, \mathrm{e})$ and monotonically increasing on $(\mathrm{e},+\infty)$. And since $\sqrt{2}\frac{1}{\sqrt[3]{3}}$, the minimum value of the sequence is $\frac{1}{\sqrt[3]{3}}$.