2002 TYS

Given that

\({{u}_{n}}={{\mathrm{e}}^{nx}}-{{\mathrm{e}}^{\left( n+1 \right)x}}\),

find \(\sum\limits_{n=1}^{N}{{{u}_{n}}}\) in terms of \(N\) and \(x\).[2]

Hence determine the set of values of \(x\) for which the infinite series

\({{u}_{1}}+{{u}_{2}}+{{u}_{3}}+...\)

is convergent and give the sum to infinity for cases where this exists.[3]