Consider the following expression :

\(\dfrac{2}{a\left(a+2\right)}=\dfrac{\left(a+2\right)-a}{a\left(a+2\right)}=\dfrac{1}{a}-\dfrac{1}{a+2}\)\(\left(a\ne0;-2\right)\)

We have :

\(P=\dfrac{2}{1.3}+\dfrac{2}{3.5}+....+\dfrac{2}{\left(2n-1\right)\left(2n+1\right)}+\dfrac{2}{\left(2n+1\right)\left(2n+3\right)}\)

\(=\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}+\dfrac{1}{2n+1}-\dfrac{1}{2n+3}\)

\(=1-\dfrac{1}{2n+3}\)

\(n>0\Rightarrow2n+3>3\Rightarrow\dfrac{-1}{3}< \dfrac{-1}{2n+3}< 0\Rightarrow\dfrac{4}{3}< 1-\dfrac{1}{2n+3}< 1\)

Hence,\(\dfrac{4}{3}< P< 1\forall n>0\)