5 It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r }\), where \(u _ { r } = x ^ { \mathrm { f } ( r ) } - x ^ { \mathrm { f } ( r + 1 ) }\) and \(x > 0\).
- Find \(S _ { n }\) in terms of \(n , x\) and the function f .
- Given that \(\mathrm { f } ( r ) = \ln r\), find the set of values of \(x\) for which the infinite series
$$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$
is convergent and give the sum to infinity when this exists.
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-10_2716_31_106_2016}
\includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-11_2723_35_101_20}
- Given instead that \(\mathrm { f } ( r ) = 2 \log _ { x } r\) where \(x \neq 1\), use standard results from the List of formulae (MF19) to find \(\sum _ { n = 1 } ^ { N } S _ { n }\) in terms of \(N\). Fully factorise your answer.