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one more thought of mine, if the series diverges, then it could be possible to reach a number.
in the case limit as n goes to infinity of 2x/x, we use l'Hôpital's rule, we take the limit and it goes to 2. This could be similar to your's, since the bottom (e^n) goes to infinity, and the top (the summation) could go to infinity (divergent). it is my experience that factorials go to infinity the fastest, as apposed to exponential, but this is when the number is not infinity...
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