Prove by mathematical induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$
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The sum \(S _ { n }\) is defined by \(S _ { n } = \sum _ { r = 1 } ^ { n } r ^ { 4 }\).
Using the identity
$$( 2 r + 1 ) ^ { 5 } - ( 2 r - 1 ) ^ { 5 } \equiv 160 r ^ { 4 } + 80 r ^ { 2 } + 2$$
show that \(S _ { n } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( 3 n ^ { 2 } + 3 n - 1 \right)\).
Find the value of \(\lim _ { n \rightarrow \infty } \left( n ^ { - 5 } S _ { n } \right)\).