Answer
Diverges
Work Step by Step
Here, we have the given series as $\Sigma_{n=1}^{\infty} \dfrac{n+1}{n (n+2)}$
Let us consider that $a_n=\Sigma_{n=1}^{\infty} \dfrac{n+1}{n (n+2)}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n}$
We can see that $a_n \leq b_n$ and $b_n$ shows a p-series with $p=1$. This implies that the series $b_n$ diverges by the p-series test.
Next, we have $\lim\limits_{n \to \infty} \dfrac{a_n}{b_n} =\lim\limits_{n \to \infty} (\dfrac{n+1}{n (n+2)}) \times n$
or, $=\lim\limits_{n \to \infty} \dfrac{n^2+n}{n^2+2n}$
or, $=\lim\limits_{n \to \infty} \dfrac{1+1/n}{1+2/n}$
or, $=1$
Hence, we can see that the given series diverges by the limit comparison test.
