2021 RI Promo Q4
A sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is defined by
${{u}_{n}}=\sum\limits_{r=1}^{n}{\left( 2r+n+1 \right)}$.
Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by ${{v}_{n}}=\frac{2}{{{u}_{n}}}$, where $n\in {{\mathbb{Z}}^{+}}$.
(i)
Find ${{u}_{n}}$ in terms of $n$.
[2]
(ii)
Show that ${{v}_{n}}=\frac{1}{n}-\frac{1}{n+1}$.
[1]
(iii)
Describe the behaviour of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .
[1]
(iv)
Find the sum, ${{S}_{N}}$, of the first $N$ terms of the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$… .
[2]
(v)
Give a reason why the series ${{S}_{N}}$ converges, and write down the value of the sum to infinity.
[1]
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