Infinite series: defining the sum Consider the following sum described below:
$$\sum_{i=0}^{x}\frac{1}{\sqrt{(3+i)(2k-4-i)}}$$
Where $0 \leq x \leq 2k-8$  and even and $k\geq 5$ is a constant integer. 
I need to find the closed form expression for this sum, however, after many attempts I couldn't. This would make it easier for me, because it makes up a function that I am proving to be increasing. Can you help me?

To show that it is increasing, consider  $$a_i=\frac1 {\sqrt{(3+i)(2k-4-i)}}$$ and let $$b_i=\frac 1 {a_i^2}=(3+i)(2k-4-i)\implies b_{i+1}-b_i=2k-2i-8$$ So  $(b_{i+1}-b_i)$ is decreasing with $i$ for a given $k$ and so $(a_{i+1}-a_i)$ is increasing.
For a closed form, I am quite skeptical (even using special functions). But, this makes a nice looking function.
