1/19 + 9/19.29 + 9/29.39 + ...+ 9/1999.2009

$\frac{1}{19}$ + $\frac{9}{19 . 29}$ + $\frac{9}{29 . 39}$ + ... + $\frac{9}{1999 . 2009}$

= $\frac{9}{9 . 19}$ + $\frac{9}{19 . 29}$ + $\frac{9}{29 . 39}$ + ... + $\frac{9}{1999 . 2009}$

= 9 · ($\frac{1}{9 . 19}$ + $\frac{1}{19 . 29}$ + $\frac{1}{29 . 39}$ + ... + $\frac{1}{1999 . 2009}$)

= $\frac{9}{10}$ · ($\frac{10}{9 . 19}$ + $\frac{10}{19 . 29}$ + $\frac{10}{29 . 39}$ + ... + $\frac{10}{1999 . 2009}$)

= $\frac{9}{10}$ · ($\frac{1}{9}$ - $\frac{1}{19}$ + $\frac{1}{19}$ - $\frac{1}{29}$ + $\frac{1}{29}$ - $\frac{1}{39}$ + ... + $\frac{1}{1999}$ - $\frac{1}{2009}$)

= $\frac{9}{10}$ · ($\frac{1}{9}$ - $\frac{1}{2009}$)

= $\frac{9}{10}$ · $\frac{2000}{18081}$

= $\frac{200}{2009}$