You have the sequence $(x_{n})_{n\in\mathbb{N}}$, so that for every $n \in \mathbb{N}$, you have to $$x_ {n}:= \ln\frac{n + 1}{n}$$ Find an explicit formula for the partial sums of the series generated by $(x_{n})_{n \in\mathbb{N}}$ and determine if the series converges.
My development I'm supposed to find a formula that generalizes the sum, so what I did is \begin{align*} S_{n}&=a_{1}+a_{2}+a_{3}+\cdot+a_{n-1}+a_{n}\\&=S_{n-1}+a_{n} \end{align*} thus, \begin{align*} a_{n}&=S_{n}-S_{n-1}\\&=\ln\frac{n+1}{n}-\ln\frac{n}{n-1}\\&=\ln\frac{(n+1)(n-1)}{n^{2}} \end{align*} and you should determine if $ \sum\limits_{n=2}^{\infty}\ln\dfrac{(n-1)(n + 1)}{n ^ {2}} $ is convergent, right?