Answer
Divergent
Work Step by Step
$$\int_{0}^{\infty }\frac{1}{\sqrt[4]{1+x}}dx=\lim_{t\rightarrow \infty}\int_{0}^{t}\frac{1}{\sqrt[4]{1+x}}dx$$
$$=\lim_{t\rightarrow \infty} \left | \frac{4}{3}(1+x)^{\frac{3}{4}} \right |_{0}^{t}$$
$$=\lim_{t\rightarrow \infty} \frac{4}{3}(1+t)^{\frac{3}{4}}-\frac{4}{3}$$
$t\rightarrow \infty \ \Rightarrow \ (1+t)^{\frac{3}{4}}\rightarrow \infty $
So the answer is divergent