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
$$T(i)=C\sum_{k=0}^{i/2}\sum_{j=0}^{k} {k \choose j} \mu^{j} \lambda^{k}=C\sum_{k=0}^{i/2} (1+\mu)^k \lambda^{k}=C\frac{\lambda(1+\mu)^{i/2+1}-1}{\lambda(1+\mu)-1}.$$
Does this help you?