By Stone-Weierstrass theorem, the set of polynomial is dense in $C[a,b]$, I am wondering what is the sequence of polynomial which can approximate absolute value function $|x|$? I know using $\sqrt{x^{2}+1/n}$ can approximate it but it is not a polynomial.
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$\begingroup$For example, you can consider the binomial expansion of $\sqrt{1-y}$ on $[0,1]$. Namely, setting $y=1-x^2$, we have$$ |x|=\sqrt{1-y}=\sum_{m=0}^{\infty}\binom{1/2}{m}(-y)^m, \quad x\in[-1,1]. $$
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