Given that ${{u}_{n}}={{\mathrm{e}}^{nx}}-{{\mathrm{e}}^{\left( n+1 \right)x}}$, find \(\sum\limit

Question

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]

Accepted Answer

$\sum_{n=1}^{N}u_n=e^x(1-e^{Nx})$ The series converges for $x\le0$, with sum $e^x$ for $x<0$ and $0$ for $x=0$.

N2002 P1 Q1 (FM)

N2002 P1 Q1 (FM)

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