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Question Number 154255 by EDWIN88 last updated on 16/Sep/21

$$\frac{5}{3^2.7^2}+\frac{9}{7^2.{11}^2}+\frac{13}{{11}^2.{15}^2}+\dots =?$$

Answered by ARUNG_Brandon_MBU last updated on 16/Sep/21

$$S=\frac{5}{3^2.7^2}+\frac{9}{7^2.{11}^2}+\frac{13}{{11}^2.{15}^2}+\dots$$$$=\sum _{n⩾1}\frac{4n+1}{{(4n-1)}^2{(4n+3)}^2}=\frac{1}{8}\sum _{n⩾1}(\frac{1}{{(4n-1)}^2}-\frac{1}{{(4n+3)}^2})$$$$=\frac{1}{8}(\frac{1}{3^2}-\frac{1}{7^2}+\frac{1}{7^2}-\frac{1}{{11}^2}+\frac{1}{{11}^2}-\frac{1}{{15}^2}+\cdot \cdot \cdot )=\frac{1}{8}\times \frac{1}{3^2}=\frac{1}{72}$$

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