ACJC/2021/JC1/Promo/P1/Q09

(a) Given that $\sum_{r=1}^n r^2=\frac{1}{6} n(n+1)(2 n+1)$, find $\sum_{r=7}^{n+1}\left(2^r+r^2-r\right)$ in terms of $n$.

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(b) (i) Use the method of differences to show that $\sum_{r=2}^n \frac{1}{r^2-1}=\frac{3}{4}+\frac{A}{n}+\frac{A}{n+1}$, where $A$ is a constant to be determined.

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(ii) Explain why the series $\sum^{\infty} \frac{1}{x^2-1}$ converges, and write down its value.

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(iii) Hence deduce that $\frac{2}{2^2}+\frac{2}{3^2}+\frac{2}{4^2}+\ldots$ is less than $\frac{3}{2}$.

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