If
${ x }_{ 1 }=\sqrt { 2 } \quad ,{ x }_{ 2 }=\sqrt { 2+\sqrt { 2 } },$ and likewise ${ x }_{ n }=\sqrt { 2+\sqrt { 2+\sqrt { 2+.......n\ terms } } }$,
then find
$2\sin\left( 1+\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +........+\frac { 1 }{ { 2 }^{ 2006 } } \right) 45°$.

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