Sequence of polynomials converging to zero function

Question

Find a sequence of polynomials $(f_n)$ such that $f_n \rightarrow 0$ point wise on $[0,1]$ and $\int_0^1 f_n(x) \rightarrow 3$. Calculate $\int_0^1 \sup_n |f_n(x)| dx$ for this sequence of polynomials.

Any help with this problem is appreciated.

Answer 1

With $f_n(x)= c_nx^n(1-x)$, we have

$$ \int_0^1f_n(x)=c_n\int_0^1x^n(1-x)\mathrm dx=\frac{c_n}{n+1}\int_0^1x^{n+1}\mathrm dx=\frac{c_n}{(n+1)(n+2)}\;, $$

so for $c_n=3(n+1)(n+2)$ the integral comes out right, and $3(n+1)(n+2)x^n(1-x)\to0$ for all $x\in[0,1]$, so the pointwise limit is $0$ on all of $[0,1]$ (including the endpoints).

P.S.: I just realized I forgot about the second part of the question; calculating that integral would be rather difficult for this sequence, and I don't immediately see how one might find a sequence for which it wouldn't be.

Answer 2

How about $f_n=c_nx^{a_n-1}(1-x)^{b_n-1}$,for some increasing positive integer sequences $\{a_n\}$,$\{b_n\}$,which satisfies $3\Gamma(a_n+b_n)=c_n\Gamma(a_n)\Gamma(b_n)$ ?