Universal Taylor series

A universal Taylor series is a formal power series $\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$, such that for every continuous function $h$ on $[-1,1]$, if $h(0)=0$, then there exists an increasing sequence $\left({\lambda }_n\right)$ of positive integers such that$$\underset{n\to \mathrm{\infty }}{\lim }\Vert \sum _{k=1}^{{\lambda }_n}{a}_k{x}^k-h(x)\Vert =0$$In other words, the set of partial sums of $\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$ is dense (in sup-norm) in $C[-1,1{]}_0$, the set of continuous functions on $[-1,1]$ that is zero at origin.

Statements and proofs

Fekete proved that a universal Taylor series exists. Template:Math proof