$$\int_{0}\limits^{\infty}\prod\limits_{k=1}^{x}\frac{\prod\limits_{n=1}^{k}\frac{\lceil k+n\rceil}{\lfloor k+n\rfloor}}{2}dx\stackrel{?}{=}\frac{3}{2}$$
There is not much context behind it. I was messing around with floor and ceiling functions when I came upon this. Desmos would not evaluate it with the integral from $0$ to $\infty$ so I plugged in $100$ instead of $\infty$ and got $1.49999999983$ which is really close to $1.5$. However, when I plug in $150$, it outputs $1.50000000011$ which is once again really close to $1.5$ but a little over this time, which I would not expect to happen if it approached $1.5$ because I had thought it would only be increasing because $\prod\limits_{k=1}^{x}\frac{\prod\limits_{n=1}^{k}\frac{\lceil k+n\rceil}{\lfloor k+n\rfloor}}{2}$ is always positive for $x>0$. I tried the upper bound of $1000$ and got $1.49999999993$ which leads me to once again believe it is converging to $1.5$. I then tried $2000$ which strangely gave me $1.50000000086$ which is further from $1.5$ than when I plugged in $150$.
Is Desmos bugging out or am I just misunderstanding something here? Also, can anyone prove or disprove this analytically? Thanks for the help in advance.
