Official solution
\[y = \left( {1 - {1 \over 2}} \right){x^2} + \left( {1 - {1 \over 3}} \right){x^3} + ....\]
\[ = ({x^2} + {x^3} + {x^4} + ......) - \left( {{{{x^2}} \over 2} + {{{x^3}} \over 3} + {{{x^4}} \over 4} + ....} \right)\]
\[ = {{{x^2}} \over {1 - x}} + x - \left( {x + {{{x^2}} \over 2} + {{{x^3}} \over 3} + ....} \right)\]
\[ = {x \over {1 - x}} + \ln (1 - x)\]
\[x = {1 \over 2} \Rightarrow y = 1 - \ln 2\]
\[{e^{1 + y}} = {e^{1 + 1 - \ln 2}}\]
\[ = {e^{2 - \ln 2}} = {{{e^2}} \over 2}\]