Thank you for clarifying the approach (and with not trying to use latex!) :).
The issue is, of course, the illegitimate manipulation of a diverging series, which was the exact issue that prompted the original article (due to Numberphile doing it) in the first place.
One thing I _will_ say for the "decaying exponent" thing is that it's somewhat similar in concept, but only somewhat, to the "smoothed sums" thing described in <https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri.... That is, if you use "e^(-x)" as your "cutoff function" and set epsilon to 1/N the two start looking quite similar.
Now e^(-x) is a totally bogus "cutoff function" per the definition in Terry's blog post, since it is not compactly supported, but it _is_ bounded, _does_ equal 1 at 0, and drops off fast enough that for practical purposes it can be used to do smoothed sums. In particular the smoothed sums will converge for most cases (e.g. anything where the sequence we're "summing" has at most polynomial growth will do so), which means you can at least try to do the rest of the analysis. I suspect, but have not checked, that the other places where compact support is used in his presentation also work out for the sorts of sequences we're talking about.
Either way, the upshot is that in some sense you have some sequence of approximations to your "actual" sum, indexed by N, and you show that for large N they all look like "power series" in 1/N which allows some finite number of negative exponents and all the approximations have matching coefficients for the negative exponents and the same constant term. And then you compute that constant term. Calling that the sum of the series is nonsense, of course, but it can still give you interesting information about something, maybe.
The issue is, of course, the illegitimate manipulation of a diverging series, which was the exact issue that prompted the original article (due to Numberphile doing it) in the first place.