5 Let $S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }$.\\
(i) Use the standard result for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ given in the List of Formulae (MF10) to show that

$$S _ { 2 n } = - n ( 2 n + 1 )$$

(ii) State the value of $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }$ and find $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q5}}