Explicit polynomial approximation of the reciprocal on the left semicircle

On $K=\{z:|z|=1,\mathrm{Re}z\le 0\}$, the polynomialsconverge uniformly to $1/z$. Expand $1/z$ first in powers of $4/(z-4)$, whose modulus is at most $4/\sqrt{17}$ on $K$, and then expand each $(z-4{)}^{-j-1}$ in powers of $z/4$. The estimate $\left(\genfrac{}{}{0ex}{}{j+k}{k}\right)\le 2^{j+k}$ controls the diagonal truncation.