KöMaL problem A, 2014

Let $n\ge1$ be a fixed integer. Calculate the distance $\inf_{p,f} \max_{0\le x\le 1} \big|f(x)-p(x)\big|$, where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x) = \sum_{k=n}^\infty c_k x^k$ defined on the closed interval $[0,1]$, where $c_k\ge0$ and $\sum_{k=n}^\infty c_k=1$.
Miklós Schweitzer competition, 2014
(5 pont)