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Ta có:
$S=\frac{3}{(1.2)^2}+\frac{5}{(2.3)^2}+\frac{7}{(3.4)^2}+\ldots+\frac{2n+1}{(n(n+1))^2}$
$=\frac{2^2-1^2}{1^2.2^2}+\frac{3^2-2^2}{2^2.3^2}+\frac{4^2-3^2}{3^2.4^2}+\ldots+\frac{(n+1)^2-n^2}{n^2.(n+1)^2}$
$=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+\ldots+\frac{1}{n^2}-\frac{1}{(n+1)^2}$
$=1-\frac{1}{(n+1)^2}$
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