Denote
$$M_i = \prod_{n=i}^K\left(1-\frac1{2\sqrt{n}}\right).$$
What could be a method of estimation of
$$\sum_{i=1}^K M_i$$ with regard to the value of $K$? It is not even of interest to find the precise explicit expression. It could be an approximate behavior with $K\rightarrow\infty$. Does it converge, diverge or decrease?
My attempts:
I've succeeded to find approximate behaviour of $M_1$ via Euler–Maclaurin formula for $\ln(M_1)$.
$$M_1 \sim \exp\left(-0.5\sqrt{K}\right)\sqrt[4]{\frac{1}{\sqrt{K}K-0.5K}}$$
But, it is not clear how to use it either intuitively to solve the problem.
Edit.
The question was deleted for some time, nevertheless Did make a solution, which is accessible here.