This question is related to [this one](https://math.stackexchange.com/questions/3382807/is-it-true-that-lim-m-to-infty-sum-k-0-fracm-12-m-choosekak). There, it is proved that
$$\forall a \in [0,1], f\_{2n+1}(a)\to\min(a,1-a), n\to\infty$$
where
$$\forall n\in\mathbb{N}, \forall a\in[0,1], f\_{2n+1}(a) :=\sum\_{k=0}^{n} {2n+1\choose{k}}\left(a^{k+1}(1-a)^{(2n+1)-k}+a^{(2n+1)-k}(1-a)^{k+1}\right).$$

However, the aforementioned result is qualitative in spirit, and now I'm looking for a quantitative version of this results. Specifically, can we show that
$$\sup\_{a\in[0,1]} |f\_{2n+1}(a)-\min(a,1-a)| \to 0, n\to +\infty?$$
If this is the case, at which rate this quantity approaches zero?