\[\boxed{\mathbf{1046}\mathbf{.}\mathbf{\ }}\]
\[\frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \frac{1}{4 \cdot 5} + .. + \frac{1}{19 \cdot 20} =\]
\[= \frac{1^{\backslash 10}}{2} - \frac{1}{20} = \frac{10}{20} - \frac{1}{20} =\]
\[= \frac{10 - 1}{20} = \frac{9}{20}\]