The sum \(S_{N}\) is defined by \(S_{N}=\sum_{n=1}^{N} n^{5}\). Using the identity
\(\left(n+\frac{1}{2}\right)^{6}-\left(n-\frac{1}{2}\right)^{6} \equiv 6 n^{5}+5 n^{3}+\frac{3}{8} n\)
find \(S_{N}\) in terms of \(N\). [You need not simplify your result.]

Hence find \(\lim _{N \rightarrow \infty} N^{-\lambda} S_{N}\), for each of the two cases
(i) \(\lambda=6\),
(ii) \(\lambda\gt 6\).