In the application of WLLN, which is the polynomial approximation. For any function $F\in C([0,1])$ can be approximated by a polynomail $G$ so as to make $||F-G||=\max_{0\le x \le 1}F(x)-G(x)$ as small as you like.
Take Bernoulli's trials with success probability $0<x<1$, introduce the polynomial $G(x)=\Bbb E[F(\frac{s_n}{n}]=\sum_{k=0}^{n}\frac{n!}{k!(n-k)!}x^k(1-x)^{n-k}F(\frac{k}{n})$. This can be proved using WLLN taking $n$ large enough $n$ that it is actually an approximation of $F$.
I am asked to verify the error for $F(x)=x$ when $x\le 1/2$ and $F(x)=1-x$ when $x>1/2$. The error of $F(1/2)-G(1/2)$ which has the form $$2\times\sum_{k<n/2}\frac{n!}{k!(n-k)!}2^{-n}(1/2-k/n)$$ is approximately $$\frac{2}{\sqrt{n}}\times \int_0^\infty \frac{e^{-x^2/2}}{\sqrt{2\pi}}xdx$$
and I notice $\int_0^\infty \frac{e^{-x^2/2}}{\sqrt{2\pi}}xdx=1$, then the error is $\frac{2}{\sqrt{n}}$.
I tried to change this sum into a Riemann sum, so I can write this as an integral. The hint says this is similar to the proof of finding the constant as $\sqrt{2\pi}$ in Stirling's approximation.
However, it is so complicated and I have no clue where to start. Could someone kindly provide some help? Thanks!
I might be wrong but it looks like you want to approximate a continuous function by Bernstein polynomials. The theory for that is well developed so it could be helpful to check that literature. One reference I like is book by Stanislaw Lojasiewicz "An Introduction to the Theory of Real Functions". There he shows for instance that (my $G_n()$ is what you call $G$) $$\|G_n(x)-F(x)\|\leq \frac{3}{2}\omega\left(\frac{1}{\sqrt{n}}\right),$$ where $\omega(\cdot)$ is the modulus of continuity of $F$ on $[0,1]$. The techniques displayed in that book could be helpful.