$\frac{1}{1.3} +\frac{1}{3.5} + \frac{1}{5.7} +...+ \frac{1}{19.21}$
=$ 1/2( \frac{2}{1.3} + \frac{2}{3.5} +\frac{2}{5.7}+...+ \frac{2}{19.21})$
= $ 1/2 . (1-1/3 + 1/3 -1/5 +1/5 -1/7 +...+1/19 -1/21 ) $
=$ 1/2 . (1- 1/21 ) $
= $ 1/2 . 20/21 = 10/21 $