How to prove this series is a telescoping series and calculate its addition

Question

In class, they have told us that we need to prove that the following series is a telescopic one by using $ln$ features and to calculate its addition.

It´s the first time we work with them, so I don´t know pretty much about them.

$$\sum_{n=2}^∞ \ln \left(1-\frac{1}{n^2}\right)$$

I also need to prove if it is convergent or not.

Thanks

Answer 1

Note that $$\sum_{n=2}^∞ \ln \left(1-\frac{1}{n^2}\right)=\sum_{n=2}^∞ \bigg[\ln \left(\frac{n+1}{n}\right)-\ln\left(\frac{n}{n-1}\right)\bigg]$$

Which is telescoping series.

Answer 2

Note that $$ 1 - \frac{1}{n^2} = \frac{n^2-1}{n} = \frac{(n+1)(n-1)}{n^2}, $$ and therefore $$ a_n = \ln\left(1 - \frac{1}{n^2}\right) = \ln \left(\frac{(n+1)(n-1)}{n^2}\right) = \ln(n+1) + \ln(n-1) - 2\ln n. $$

Can you now prove that $\sum a_n$ is telescoping?