Given that \(\displaystyle \sum_{r=1}^{n}\frac{2}{(r+4)(r+6)}=\frac{11}{30}-\frac{1}{n+5}-\frac

Question

Given that $\displaystyle \sum_{r=1}^{n}\frac{2}{(r+4)(r+6)}=\frac{11}{30}-\frac{1}{n+5}-\frac{1}{n+6}$, Find $\displaystyle \sum_{r=5}^{n+4}\frac{2}{r(r+2)}$.[2] Hence, show that $\displaystyle \sum_{r=5}^{n+4}\frac{1}{(r+1)^2}<\frac{11}{60}$.[3]

Accepted Answer

(i) $\frac{11}{30}-\frac{1}{n+5}-\frac{1}{n+6}$ (ii) $\displaystyle \sum_{r=5}^{n+4}\frac{1}{(r+1)^2}<\frac{11}{60}$

Reindexed rational sum and an upper bound

Reindexed rational sum and an upper bound

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