\frac{1}{1\cdot 3}+\frac{1}{3\cdot 5}+\frac{1}{5\cdot 7}+\:.\:.\:.\:\:+\frac{1}{\left(2n-1\right)\left(2n+1\right)}=\left(\frac{1}{2n+1}\right)
Mathbot Says...
I wasn't able to parse your question, but the HE.NET team is hard at work making me smarter.