$A = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + .... + \frac{1}{{1024}}$ $2 \times A = \frac{2}{2} + \frac{2}{4} + \frac{2}{8} + \frac{2}{{16}} + \frac{2}{{32}} + .... + \frac{2}{{1024}}$ $2 \times A = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .... + \frac{1}{{512}}$ $2 \times A - A = \left( {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .... + \frac{1}{{512}}} \right) - \left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + ... + \frac{1}{{1024}}} \right)$ $A = 1 - \frac{1}{{1024}} = \frac{{1023}}{{1024}}$