Answer
Diverges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{(-2)^{n}}{\sqrt{n}}$$
By using the Ratio Test, we get:
\begin{aligned} \rho &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \\
&=\lim _{n \rightarrow \infty} \left|\frac{(-2)^{n+1}}{\sqrt{n+1}}\frac{\sqrt{n}} {(-2)^{n}}\right| \\
&=2\lim _{n \rightarrow \infty} \left|\frac{\sqrt{n}}{\sqrt{n+1}} \right| \\
&=2>1 \end{aligned}
Thus the series diverges.
