Sequence and series

2013 RI P1 Q8 [Modified] (i) Show that ${{sin }^{4}}A={{sin }^{2}}A-frac{1}{4}{{sin }^{2}}2A$. It is given that ${{S}_{n}}=sumlimits_{r=0}^{n}{frac{1}{{{4}^{r}}}{{sin }^{4}}left[ {{2}^{r}}left( frac{pi }{3} right) right]}$. (ii) By using the result in (i),

NJC Sigma Notation Tutorial Q2 (i) Prove that for all positive integers $n$, $frac{1}{n!}-frac{1}{left( n+1 right)!}=frac{1}{n!+left( n-1 right)!}$. (ii) Hence evaluate $sumlimits_{n=1}^{N}{frac{1}{n!+left( n-1 right)!}}$ in terms of $N$. (iii) By

NJC Sigma Notation Tutorial Q1 The method of differences can be used to evaluate $sumlimits_{r=1}^{n}{text{f}left( r right)}$, where $text{f}left( r right)=frac{2}{r}-frac{3}{r+1}+frac{1}{r+2}$. In each of the following situations, explain clearly the

Application of Sum to Infinity Q3 (i) (i) By considering $text{f}left( r right)-text{f}left( r+1 right)$, where $text{f}left( r right)=frac{1}{r}$, find the sum to $2n$ terms of the series $frac{1}{2}+frac{1}{6}+frac{1}{12}+frac{1}{20}+…$ (ii)

Sigma Notation Tutorial Q1 (i) Use the method of differences to show that $sumlimits_{r=2}^{n}{frac{1}{{{r}^{2}}-1}}=frac{3}{4}+frac{A}{n}+frac{A}{n+1}$, where $A$ is a constant to be determined. [3] (ii) Find $sumlimits_{r=n+1}^{2n}{frac{1}{{{r}^{2}}-1}}$. [2] (iii) Explain why