2002 TYS

Given that

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

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]