Sigma notation: changing the index

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">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.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find $\sum\limits_{r=n+1}^{2n}{\frac{1}{{{r}^{2}}-1}}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Explain why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{2}}-1}}$ converges, and write down its value.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence deduce that $\frac{2}{{{2}^{2}}}+\frac{2}{{{3}^{2}}}+\frac{2}{{{4}^{2}}}+...$ is less than $\frac{3}{2}$.</span></div><p style="text-align: right;">[2]</p></li></ol>

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.

Find $\sum\limits_{r=n+1}^{2n}{\frac{1}{{{r}^{2}}-1}}$.

Explain why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{2}}-1}}$ converges, and write down its value.

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

Sigma Notation Tutorial Q1

Sigma Notation Tutorial Q1

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