A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{1}}=1$ and ${{u}_{n+1}}={{u}

Question

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{1}}=1$ and ${{u}_{n+1}}={{u}_{n}}-\frac{2n+1}{{{n}^{2}}{{\left( n+1 \right)}^{2}}}$, for all $n\ge 1$. Given that ${{u}_{n}}=\frac{1}{{{n}^{2}}}$, find $\sum\limits_{n=1}^{N}{\frac{2n+1}{{{n}^{2}}{{\left( n+1 \right)}^{2}}}}$. [2] Give a reason why the series in part (i) is convergent and state the sum to infinity. [2] Use your answer to part (i) to find $\sum\limits_{n=2}^{N}{\frac{2n-1}{{{n}^{2}}{{\left( n-1 \right)}^{2}}}}$. [2]

Accepted Answer

(i) $1-\frac1{(N+1)^2}$; (ii) convergent with sum $1$; (iii) $1-\frac1{N^2}$

N2007 P2 Q2 [Modified]

N2007 P2 Q2 [Modified]

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