#### Answer

Convergent and converge to $1$.

#### Work Step by Step

We have $_n=\Sigma_{n=1}^{\infty} \dfrac{1}{k (k+1)}=\Sigma_{n=1}^{\infty} \dfrac{1}{k }-\dfrac{1}{{(k+1)}}$
This gives that $s_n=1-\dfrac{1}{n+1}$
Thus, $\lim\limits_{n \to \infty} s_n= \lim\limits_{n \to \infty} (1-\dfrac{1}{n+1})=1$
Now, $T_{2n+1}=T_{2n}+\dfrac{1}{n+1}=(1-\dfrac{1}{n+1})+\dfrac{1}{n+1}=1$
Thus, $\lim\limits_{n \to \infty} s_n= \lim\limits_{n \to \infty} (1-\dfrac{1}{n+1})=1$
So, both series are convergent and they converge to $1$.