Sigma Notation Tutorial Q1
(i)
Use the method of differences to show that $\sum\limits_{r=2}^{n}{\frac{1}{{{r}^{2}}-1}}=\frac{3}{4}+\frac{A}{n}+\frac{A}{n+1}$, where $A$ is a constant to be determined.
[3]
(ii)
Find $\sum\limits_{r=n+1}^{2n}{\frac{1}{{{r}^{2}}-1}}$.
[2]
(iii)
Explain why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{2}}-1}}$ converges, and write down its value.
[2]
(iv)
Hence deduce that $\frac{2}{{{2}^{2}}}+\frac{2}{{{3}^{2}}}+\frac{2}{{{4}^{2}}}+…$ is less than $\frac{3}{2}$.
[2]
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