Algebra Level 5

For the given nested radical, a real number L exists as the evaluation of the expression.

$$2\cdot 2\sqrt { 2\sqrt [ 3 ]{ 2\sqrt [ 5 ]{ 2... } } }$$ = L

If the sum of the first $$\left\lceil L \right\rceil$$ terms of the decimal expansion of L is S, then what is S?

Assumptions:

The nth radical index is the nth fibonacci number. The first two numbers can be written as numbers with a radical of index 1.

Inspired by Brian Charlesworth's problem Harder Nested Radicals

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