We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. Deduce that, for $t\in\left[0,1\left[$ and $n\in\mathbb{N}$, $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.
We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Deduce that, for $t\in\left[0,1\left[$ and $n\in\mathbb{N}$, $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.