What an Amusing Nested Radical!

Calculus Level 3

\[\Large \sqrt{2\sqrt[3]{2\sqrt[4]{2\sqrt[5]{2\cdots}}}}\]

Let the value of the above expression be denoted by \(I\). Find \(\lfloor1000I\rfloor\).

Details and Assumptions:

You may use a scientific calculator to evaluate the final result.

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