Given that  $$\left( 1+\frac{1}{2} \right)\times \left( 1+\frac{1}{{{2}^{2}}} \right)\times \left( 1+\frac{1}{{{2}^{4}}} \right)\times \left( 1+\frac{1}{{{2}^{8}}} \right)\times \cdots \times \left( 1+\frac{1}{{{2}^{{{2}^{2009}}}}} \right)$$$$=2\left( 1-\frac{1}{{{2}^{{{2}^{n}}}}} \right)$$,  find the value of $$n$$. 
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