Show that $\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}=\frac{A}{{{n}^{3}}-n}$, where $A$ is a constant t

Question

Show that $\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}=\frac{A}{{{n}^{3}}-n}$, where $A$ is a constant to be found. [2] Hence find $\sum\limits_{r=2}^{n}{\frac{1}{{{r}^{3}}-r}}$, (There is no need to express your answer as a single algebraic fraction.) [3] Give a reason why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{3}}-r}}$ converges, and write down its value. [2]

Accepted Answer

(i) $A=2$; (ii) $\frac12\left(\frac12-\frac1n+\frac1{n+1}\right)$; (iii) $\frac14$

N2009 P1 Q3

N2009 P1 Q3

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