Alternating series summation

1 Find \(2 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }\). Hence find \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 n ) ^ { 2 }\), simplifying your answer.

5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).

1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).

5 Let $$S _ { N } = \sum _ { n = 1 } ^ { N } ( - 1 ) ^ { n - 1 } n ^ { 3 }$$ Find \(S _ { 2 N }\) in terms of \(N\), simplifying your answer as far as possible. Hence write down an expression for \(S _ { 2 N + 1 }\) and find the limit, as \(N \rightarrow \infty\), of \(\frac { S _ { 2 N + 1 } } { N ^ { 3 } }\).

Show that \(\left( n + \frac { 1 } { 2 } \right) ^ { 3 } - \left( n - \frac { 1 } { 2 } \right) ^ { 3 } \equiv 3 n ^ { 2 } + \frac { 1 } { 4 }\). Use this result to prove that \(\sum _ { n = 1 } ^ { N } n ^ { 2 } = \frac { 1 } { 6 } N ( N + 1 ) ( 2 N + 1 )\). The sums \(S , T\) and \(U\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } , \\ & T = 1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 2 N - 1 ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } , \\ & U = 1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } . \end{aligned}$$ Find and simplify expressions in terms of \(N\) for each of \(S , T\) and \(U\). Hence

1. describe the behaviour of \(\frac { S } { T }\) as \(N \rightarrow \infty\),
2. prove that if \(\frac { S } { U }\) is an integer then \(\frac { T } { U }\) is an integer.

10. Gareth is investigating a series involving cube numbers. His series is $$1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 } - 6 ^ { 3 } + 7 ^ { 3 } - \ldots$$ Gareth continues his series and ends with an odd number.
Find and simplify an expression for the sum of Gareth's series in terms of \(k\), where \(k\) is the number of odd numbers in his series.