Given that \(\sum\limits_{r=2}^{n}{\frac{2}{(r+3)(r+5)}}=\frac{11}{30}-\frac{2n+9}{\left( n+4 \

Question

Given that $\sum\limits_{r=2}^{n}{\frac{2}{(r+3)(r+5)}}=\frac{11}{30}-\frac{2n+9}{\left( n+4 \right)\left( n+5 \right)}$. Hence find $\sum\limits_{r=4}^{n+4}{\frac{2}{r(r+2)}}$. Deduce that $\sum\limits_{r=4}^{n+4}{\frac{1}{{{(r+1)}^{2}}}}<\frac{9}{40}$.

Accepted Answer

(ii) $\frac{9}{20}-\frac{2n+11}{\left( n+5 \right)\left( n+6 \right)}$ (iii) $\displaystyle\sum_{r=4}^{n+4}\frac{1}{(r+1)^2} < \frac{9}{40}$

2015 AJC P1 Q8 [Modified]

2015 AJC P1 Q8 [Modified]

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