Limit of ratio of sums

4 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

1. \(\lambda = 6\),
2. \(\lambda > 6\).

4 In this question you must show detailed reasoning.
Determine the smallest value of \(n\) for which \(\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341\).