Consider the following integral
$$\int_0^1 (1-x^n)^M \,d x$$
It converges to $0$ as $M\to\infty$, but I would like to find bounds on the convergence rates. What I mean is that it is straightforward to find constants A and B such that
$$\frac{A}{M}<\int_0^1 (1-x^n)^M \,d x<\frac{B}{M^{1/n}}$$
However, is it possible to obtain identical upper and lower bounds? The claim that I have is that the lower bound can also be made of the form $A'/M^{1/n}$, but I have not been able to prove it.