Solving the Summation Puzzle: Analyzing Convergence and Calculating the Sum

neom
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\sum^k_{n=1}e^{-n\sum^k_{n=2}...e^{-n\sum^k_{n=k-1}e^{-n}}}

Can anyone help me find out if this converges and if so how to calculate the sum?
I don't have an idea on how to even start.

This is not homework
 
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