Okay so here I go again studying summability theory I was wondering the following problem but first I'll state a few conventions:
A series diverges if the partial sums tends to $\pm \infty$, converges if the partial sums tends to a finite limit (the series will be called summable with sum such limit) and otherwise is called oscillating. Now in the context of summability theory I say that a series (convergent or not) is resummable with the method $\mathcal{M}$ if such method can assign a finite value to it. Everything so far is pretty standard but I prefer to clarify that I'm not working in the "standard context" of convergent series but rather with generalized sums and limits.
Now doing some calculations one can come up with divergent series of real numbers which can be resummed to complex values for example:
$$\sum_{n=0}^{+\infty}\frac {(2n)!}{(n+1)!n!}=1+1+2+10+\dots=\frac 12-\frac{\sqrt{3}}{2}i$$
however doing various example I've noticed that only divergent series happens to be resummed to complex values, oscillating series always are resummed to real values! This sounds pretty odd to me as there are a lot of different summability method and this would be a statement too general (and maybe too good) to be true imho.
So my question is: can someone come up with and example of an oscillating series of real numbers which can be resummed by some (more or less explicit) summability method to a complex value? Maybe I'm missing something very trivial, I apologize in advance if that's the case.
edit: for the one curious which method I used to sum the series I've simply applied Abel's method related to analytic continuation:
$$\sum_{n=0}^{+\infty}\frac {(2n)!}{(n+1)!n!}=\lim_{x\to 1^-}\sum_{n=0}^{+\infty}\frac {(2n)!}{(n+1)!n!}x^n=\lim_{x\to 1^-} \frac {1-\sqrt{1-4x}}{2x}$$