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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Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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