Problem - 4568
Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$
$$0\le \lim_{n\to\infty}\int_0^1x^n(1-x)^n dx \le \lim_{n\to\infty}\int_0^1 \left(\frac{x + (1-x)}{2}\right)^ndx=0 $$
Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$
$$0\le \lim_{n\to\infty}\int_0^1x^n(1-x)^n dx \le \lim_{n\to\infty}\int_0^1 \left(\frac{x + (1-x)}{2}\right)^ndx=0 $$