Let \(u_r = e^{rx}(e^{2x} - 2e^x + 1)\).

(a) Using the method of differences, or otherwise, find \(\sum_{r=1}^{n} u_r\) in terms of \(n\) and \(x\).

(b) Deduce the set of non-zero 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.

(c) Using a standard result from the list of formulae (MF19), find \(\sum_{r=1}^{n} \ln u_r\) in terms of \(n\) and \(x\).