I have a infinite sum that looks like the following:
$$ \sum_{k = 0}^{\infty} \left(\frac{1}{1 + k/A}\right)^a \frac{(-i B)^k}{k!} $$
Here, A is a positive real number and $a$ is a positive integer. The $i$ is the imaginary unit $i = \sqrt{-1}$. Can I find an asymptotic or a an integer $K$ for which:
$$ \sum_{k = K+1}^{\infty} \left(\frac{1}{1 + k/A}\right)^a \frac{(-i B)^k}{k!} < \epsilon $$
Yes. You can estimate $$\sum_{k=0}^\infty\left\vert\left(\frac{1}{1+k/A}\right)^a\frac{(-iB)^k}{k!}\right\vert\leq\sum_{k=0}^\infty\frac{|B|^k}{k!}=e^{|B|}.$$ Thus, the series converges (absolutely) and you can consequently find such a $K$.