Infinite Infinities

Algebra Level 3

\[ 2\sum _{ n=0 }^{ \infty }{ { \left[ { 2\left[ \frac { 2 \sum _{ n=0 }^{ \infty }{ { \left[ { 2\left[ \frac { 2 \sum _{ n=0 }^{ \infty }{ { \left[ { 2\left[ \frac { ... }{ 2013 } \right] }^{ n } \right] }^{ -1 } } }{ 2013 } \right] }^{ n } \right] }^{ -1 } } }{ 2013 } \right] }^{ n } \right] }^{ -1 } } \]

What is the value of the nested summation above?


This might look daunting at first but the solution will come out very nicely and easily.
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