Sigma Notation

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

2017 RVHS Promo Q5 A sequence ${{U}_{1}}$, ${{U}_{2}}$, ${{U}_{3}}$,… is defined by ${{U}_{n}}=frac{n}{{{text{e}}^{n}}}$. (i) Show that $frac{nleft( 1-text{e} right)+1}{{{text{e}}^{n+1}}}={{U}_{n+1}}-{{U}_{n}}$ . [1] (ii) Hence find $sumlimits_{r=1}^{n}{left( frac{rleft( 1-text{e} right)+1}{{{text{e}}^{r+1}}} right)}$ in

2017 NYJC Promo Q5 (i) By considering $tan left( A-B right)$ or otherwise, show that   ${{tan }^{-1}}left( k+1 right)-{{tan }^{-1}}left( k right)={{cot }^{-1}}left( {{k}^{2}}+k+1 right)$ for $k>0$. [2]   (ii)

2021 RI Promo Q4 A sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is defined by ${{u}_{n}}=sumlimits_{r=1}^{n}{left( 2r+n+1 right)}$. Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by ${{v}_{n}}=frac{2}{{{u}_{n}}}$, where $nin {{mathbb{Z}}^{+}}$. (i) Find

2022 CJC Promo Q8 (i) Show that $frac{1}{r!}-frac{1}{left( r+1 right)!}=frac{r}{left( r+1 right)!}$. [1] (ii) Hence find $sumlimits_{r=1}^{N}{frac{r}{left( r+1 right)!}}$ in terms of $N$. [3] (iii) Explain why the series in

2022 NYJC Promo Q6 A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$,… is such that ${{u}_{r}}=frac{1}{r}-frac{1}{r+2}$ where $rge 1$. (i) Show that ${{u}_{r}}=frac{A}{rleft( r+2 right)}$ for a constant $A$ to be determined. Describe