Train of thought
In computer science, for representing signed integers, we can use the two's complement representation, so that $11111111_{(2)} = -1$ over 8 bits with $x_{(2)}$ denoting binary representation. This is used because $11111111_{(2)} + 00000001_{(2)} = 00000000_{(2)}$ (the overflowed 1 go to the non-existing ninth bit).
This made me think a bit about the mathematical aspect of this relation. Given the divergent series (representing an infinity of ones in base 2): $$S = \sum_{i=0}^\infty2^i$$ If we wanted to assign it a constant value, one can find that: $$2\times S = \sum_{i=0}^\infty2^{i+1} = \sum_{i=1}^\infty2^i = S - 2^0$$ $$S = -1$$ Which is the result used in computer science. So, in a way, with only natural positive numbers, we get negative numbers.
The question
Is there a way to construct a divergent series with elements in $\mathbb{Z}$ such that with a specific summation method, the result would be a imaginary number? If not, prove it.