Find $\sum\limits_{r=N+1}^{2N}{\left( {{2}^{r}}+{{3}^{N}} \right)}$.[3] The \({{n}^{\mathrm{th}}}\

Question

Find $\sum\limits_{r=N+1}^{2N}{\left( {{2}^{r}}+{{3}^{N}} \right)}$.[3] The ${{n}^{\mathrm{th}}}$ term, ${{u}_{n}}$, of a series is given by ${{u}_{n}}=\frac{1}{{{n}^{2}}+n}-\frac{1}{{{n}^{2}}+3n+2}$. Find $\sum\limits_{n=1}^{N}{{{u}_{n}}}$ in terms of $N$. Explain why $\sum\limits_{n=1}^{\infty }{{{u}_{n}}}$ is a convergent series, and state the value of the sum to infinity.[5]

Accepted Answer

(a) ${{2}^{N+1}}\left( {{2}^{N}}-1 ight)+N{{3}^{N}}$ (b) $\sum\limits_{n=1}^{\infty }{{{u}_{n}}}$is a convergent series 	ext{ and } the sum to infinity $=\frac{1}{2}$.

2018 RI Promo Q6

2018 RI Promo Q6

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