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Question Number 81507 by Power last updated on 13/Feb/20

Commented by mr W last updated on 13/Feb/20

$$wehave\frac{2n+1}{n^2{(n+1)}^2}=\frac{1}{n^2}-\frac{1}{{(n+1)}^2}$$$$$$$$\frac{3}{{\left(1\times 2\right)}^2}+\frac{5}{{\left(2\times 3\right)}^2}+...+\frac{19}{{\left(9\times 10\right)}^2}$$$$=(\frac{1}{1^2}-\frac{1}{2^2})+(\frac{1}{2^2}-\frac{1}{3^3})+...+(\frac{1}{9^2}-\frac{1}{{10}^2})$$$$=1-\frac{1}{{10}^2}$$$$=\frac{99}{100}$$

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