Answer
Diverges.
Work Step by Step
Given $$ \frac{0}{1}-\frac{1}{2}+\frac{2}{3}-\frac{3}{4}+\cdots= \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n-1)}{n}$$
Since
\begin{align*}
\lim _{n \rightarrow \infty} \frac{(-1)^{n-1}(n-1)}{n}&=\lim _{n \rightarrow \infty} (-1)^{n-1}(1-1/n) \\
&=\begin{cases}
1&n\ \text{even}\\
-1&n\ \text{odd}
\end{cases}
\end{align*}
Since the $n$th term $a_{n}$ does not converge to zero, thus the series diverges.
