How to adjust the base for a super-exponential function?

I have the following function:

$$
T(i) = (sum_{k=0}^{i/2} sum_{j=0}^k binom k j mu^j lambda^k ) C
$$

where $C$ is a constant. This function is diverging for certain values of $mu$ and $lambda$ (such as $mu = lambda = 0.75$), and converging for certain other values (such as $mu = lambda = 0.5$). I can see that by simulating the function for different values of $mu$ and $lambda$.

My question is whether and how I could calculate the lowest $mu$, $lambda$, or $mu times lambda$ value that would make this function diverging, or the highest $mu$, $lambda$, or $mu times lambda$ that would make this function converging.

Thanks so much!

Regards,

Utku