Given a positive integer $n$, consider $f_n = -min_{|z|=1} Re sum_{i>n} frac{z^i e^{-i/n}}{i}$. What can be said about the growth of $f_n$? How large can it get?

Taking maximum instead of minimum, the answer is obvious as the maximum is attained at $z=1$ and this becomes a real analysis problem (which in particular implies $f_n$ is bounded, but perhaps $f_n$ tends to $0$ at some rate).

An approach suggested by a friend is to express $sum_{i>n} t^i/i$ in terms of a Cauchy integral related to $-ln (1-zeta t)$ and then perhaps apply a saddle point estimate (uniformly in $|t|=e^{-1/n}$) — but I wasn’t able to carry this out.