Answer
Diverges.
Work Step by Step
From the Definition of Improper Integrals with Infinite Integration Limits
Case 1.
If $f$ is continuous on the interval $[a, \infty)$, then $\displaystyle \int_{a}^{\infty}f(x)d\mathrm{x}=\lim_{b\rightarrow\infty}\int_{a}^{b}f(x)dx$.
---
$\displaystyle \frac{4}{\sqrt[4]{x}}$ is continuous on $[a, \infty)$.
$\displaystyle \int_{1}^{\infty}\frac{4}{\sqrt[4]{x}}dx=\lim_{b\rightarrow\infty}\int_{1}^{b}4x^{-1/4}dx$
$=\displaystyle \lim_{b\rightarrow\infty}[\frac{16}{3}x^{3/4}]_{1}^{b}$
$=\displaystyle \lim_{b\rightarrow\infty}[\frac{16}{3}x^{3/4}]_{1}^{b}$
$=\infty -\displaystyle \frac{16}{3}$
$=\infty$
Diverges.
