Answer
$$
\sum_{n=1}^{\infty} \frac{1}{n^{\frac{1}{4}}}
$$
Apply the Integral Test to the given series we obtain
$$
\int_{1}^{\infty} \frac{1}{x^{\frac{1}{4}}} d x = \infty
$$
So, the series diverges
Work Step by Step
$$
\sum_{n=1}^{\infty} \frac{1}{n^{\frac{1}{4}}}
$$
Apply the Integral Test to the series
$$
\sum_{n=1}^{\infty} \frac{1}{n^{\frac{1}{4}}}
$$
The function
$$
f(x)=\frac{1}{x^{\frac{1}{4}}}
$$
is positive and continuous for $x \geq 1$ . To determine whether $f$ is decreasing, find the derivative.
$$
f^{'}(x)=\frac{-1}{4x^{\frac{5}{4}}}
$$
So, $f^{'}(x) \lt 0 $ for $x\geq 1$,
and it follows that $f$ satisfies the conditions for the Integral Test.
You can integrate to obtain
$$
\begin{aligned}
\int_{1}^{\infty} \frac{1}{x^{\frac{1}{4}}} d x &=\lim\limits_{t \to \infty } \int_{1}^{t } \frac{1}{x^{\frac{1}{4}}} d x \\
&=\lim\limits_{t \to \infty }[\frac{4}{3}x^{\frac{3}{4}} ]_{1}^{t}\\
&=[\frac{4}{3}x^{\frac{3}{4}} ]_{1}^{\infty} \\
&=\infty
\end{aligned}
$$
So, the series diverges