\(\mathbf{1}\) Verify that \(\frac{1}{n^{2}}-\frac{1}{(n+1)^{2}}=\frac{2 n+1}{n^{2}(n+1)^{2}}\).
Let \(S_{N}=\sum_{r=1}^{N} \frac{2 r+1}{r^{2}(r+1)^{2}}\). Express \(S_{N}\) in terms of \(N\).
Let \(S=\lim _{N \rightarrow \infty} S_{N}\). Find the least value of \(N\) such that \(S-S_{N}\lt 10^{-16}\).