Express $\frac{2}{\left( r-1 \right)r\left( r+1 \right)}$ in partial fractions.[2] Hence find, in

Question

Express $\frac{2}{\left( r-1 \right)r\left( r+1 \right)}$ in partial fractions.[2] Hence find, in terms of $n$, $\sum\limits_{r=2}^{n}{\frac{2}{\left( {{r}^{2}}-1 \right)r}}$ .[3] Using the result in part (b), show that $\sum\limits_{r=2}^{n}{\frac{1}{{{r}^{3}}}<\frac{1}{4}}$.[3]

Accepted Answer

(a) $\frac{1}{r-1} - \frac{2}{r} + \frac{1}{r+1}$; (b) $\frac{1}{2} - \frac{1}{n} + \frac{1}{n+1}$; (c) $\sum_{r=2}^{\infty} \frac{1}{r^3} < \frac{1}{4}$

2024 CJC MYE P1 Q7

2024 CJC MYE P1 Q7

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