Official solution
1 + \[{3 \over 2}\] + \[{7 \over 4}\] + \[{15 \over 8}\] + \[{31 \over 16}\] + . . . .
= (2 - 1) + (2 - \[{1 \over 2}\] ) + (2 - \[{1 \over 4}\]) + (2 - \[{1 \over 8}\]) + . . . . .+ 20 terms
= (2 + 2 + . . . . . 20 terms) - (1 + \[{1 \over 2}\] + \[{1 \over 4}\] + . . . . . 20 terms)
= 2 \[ \times \] 20 - \[\left( {{{1 - {{\left( {{1 \over 2}} \right)}^{20}}} \over {1 - {1 \over 2}}}} \right)\]
= 40 - 2 + 2 \[{\left( {{1 \over 2}} \right)^{20}}\]
= 38 + \[{1 \over {{2^{19}}}}\]