Let $f \in \mathcal{H}(U)$ a holomorphic function, where $U\subset X$ is an open balanced set in an infinite dimensional Banach space $X$, with power series around $0$
$$f=\sum_{n=0}^\infty P_n,$$
and let $K\subset U$ be a compact set. Suppose that 
$$\sum_{n=0}^\infty \|P_n\|_K < \delta.$$
I want to show that there is a $\varepsilon>0$ such that
$$\sum_{n=0}^\infty \|P_n\|_{K+B_\varepsilon}<\delta.$$ 
This problem is connected with the next question: https://mathoverflow.net/questions/431771/seminorms-ported-by-a-compact .Here, I showed that the series is always finite.