Ta có:
$A = \dfrac{1}{1.3} +\dfrac{1}{3.5} +\dots +\dfrac{1}{(2n-1)(2n+1)}$
$\to 2A = \dfrac{2}{1.3} +\dfrac{2}{3.5} +\dots +\dfrac{2}{(2n-1)(2n+1)}$
$\to 2A =\dfrac{3-1}{1.3} +\dfrac{5-3}{3.5} +\dots +\dfrac{2n+1 - (2n-1)}{(2n-1)(2n+1)}$
$\to 2A =\dfrac{3}{1.3} -\dfrac{1}{1.3}+\dfrac{5}{3.5}-\dfrac{3}{3.5} +\dots +\dfrac{2n+1}{(2n-1)(2n+1)}-\dfrac{2n-1}{(2n-1)(2n+1)}$
$\to 2A =1 -\dfrac{1}{2n+1}$
$\to A =\dfrac12 -\dfrac{1}{2(2n+1)}$
$\to A <\dfrac12\quad \forall n \geq 1$
