N2007 P2 Q2

A sequence \(u_1,u_2,u_3,\ldots\) is such that \(u_1=1\) and \(u_{n+1}=u_n-\frac{2n+1}{n^2(n+1)^2}\), for all \(n\geq1\).

1. Use the method of mathematical induction to prove that \(u_n=\frac1{n^2}\).[4]
2. Hence find \(\sum_{n=1}^N\frac{2n+1}{n^2(n+1)^2}\).[2]
3. Give a reason why the series in part (ii) is convergent and state the sum to infinity.[2]
4. Use your answer to part (ii) to find \(\sum_{n=2}^N\frac{2n-1}{n^2(n-1)^2}\).[2]