Consider the sequence of functions $ f_n : [0,1] \to \mathbb{R} $ given by
$$ f_n(x) = (n+1)(n+2)(1-x)x^n. $$
Calculate:
$$ \int \lim_{n \to \infty} f_n(x) \, d\mu_L. $$
It is evident to me, that this sequences becomes zero. However i am not being able to show this statement cleanly. I have tried rewriting the function as $$f_n(x)= (\partial^2/\partial x^2 x^{n+2}) \cdot(1-x) $$ And in this form, it is evident, that the limit over n only effects the first part, however ,i have no mechanism to switch the limit with the partial derivative.
And in a differnt note, can one find a lesbegue integrable function $g$ such that $$\vert f_n(x)\vert \leq g$$ in order to apply the theorem of Lesbegue? The Integral of the function itself is not hard to calculate, and then apply the limit to the result.
