First we can see that ${\frac{(6k+1)^{k}}{(2k+5)^{k}}}$ behaves asymptotically as $3^k$, while $(z-2i)^k$ is divergent if $\|z-2i\|\ge1 $ otherwise it is convergent. On the other hand we can extend this argument to the full sequence, so we need only  $\|z-2i\| < \frac{1}{3} $ to ensure that the sequence is convergent, and in fact _Mathematica_ can tackle  with an approprate assumption:

    Limit[ ((6k + 1)/(2k + 5))^k (z - 2I)^k, k -> Infinity, 
            Assumptions -> 3 Abs[-2 I + z] < 1]
>     0

Moreover it can be extended to the adequate series, this provides the condition for convergence of the series:

    SumConvergence[((6k + 1)/(2k + 5))^k (z - 2I)^k, k]
>     3 Abs[-2 I + z] < 1


_Mathematica_ correctly returned `ComplexInfinity` for provided values of `z` since they did not satisfied the appropiate condition:

    Abs[{3 + 4I, 3 - 4I, -3 - 4I, -4I, 4I, 4, -4}]
>     {5, 5, 5, 4, 4, 4, 4}
 
Concluding we can say that _Mathematica_ still requires a quantum of mathematical knowledge and I think it is quite fine. I'm quite convinced that the perspective that in the distant future mathematics will be done by computers is only a unrealizable dream (and some kind of S.Wolfram's publicity). See [Computational Knowledge and the Future of Pure Mathematics](http://blog.wolfram.com/2014/08/12/computational-knowledge-and-the-future-of-pure-mathematics/). Nevertheless we can still expect quite reasonably that _Mathematica_ will be able to do much more in mathematics.