(Solved): Using the definition show that the following sequence is not a Cauchy sequence: \[ x_{n}=\frac{1}{ ...

Using the definition show that the following sequence is not a Cauchy sequence: \[ x_{n}=\frac{1}{2^{2}}+\frac{2}{3^{2}}+\ldots+\frac{n}{(n+1)^{2}} \] [Hint: consider \( x_{2 n}-x_{n} \). You can use the fact that if \( \left\{x_{n}\right\} \) is Cauchy then \( x_{2 n}-x_{n} \) must converge to 0 , we proved it in class.]