Consider the series $\sum_{k=1}^n k e^{-\varepsilon k}\cos(\varepsilon k)$. If one lets $\varepsilon$ be small enough (for example 0.0001) and $n$ large enough (for example 1.000.000), one sees by numerical computations that this sum will converge to $-\frac{1}{12}$. See also https://en.wikiversity.org/wiki/MATLAB/Divergent_series_investigations or test it by yourself using any programming language.
Besides this rather interesting fact, my question is now how to solve $\varepsilon$ in terms of $n$ in the following equation $$\sum_{k=1}^n k e^{-\varepsilon k}\cos(\varepsilon k)=-\frac{1}{12}.$$ If one solves this, one can for example obtain sufficient conditions in order to apply zeta function regularization in areas where is made a lot of use of perturbed series (such as physics).
I tried to integrate both sides of the expression (w.r.t. $\varepsilon$) but arrived at no interesting relationships so far. Maybe one does have heard of this expression, knows more about it or even know how to solve it for $\varepsilon$?
See also my slightly related question Closed form expression for (periodic) generalized harmonic numbers? which might provide clearer insights.