Answer
Divergent
Work Step by Step
It is given a series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\ldots$.
The $n-$term of the series is $a_n=\frac{n}{n+1}$.
Find the limit of $a_n$:
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{n}{n+1}=\lim\limits_{n \to \infty}\frac{1}{1+\frac{1}{n}}=\frac{1}{1+0}=1\neq 0$
Since $\lim\limits_{n \to \infty}a_n\neq 0$, it follows by The Test for Divergence that the series is divergent.
