I want to analyze the $1-\alpha$-quantile, $\alpha\in(0,1)$, of a $F_{n, m}$ distribution, keeping n fixed while increasing m. It seems that the quantile decreases monotonically, but I would like to be sure about this and furthermore know if it converges to a specific value.

I am aware that the cumulative distribution function is given by an incomplete beta function. However, inverting it to get the quantile function does not seem trivial. Are there existing results on this?