Let $X$ be a Banach space and let $T_n\colon X\to X$ be a family of bounded operators convergent to some operator $T\colon X\to X$. Is it true that
$T(X)\subseteq \sum_{n=1}^\infty T_n(X)$?
I mean by $\sum_{n=1}^\infty V_n$ the set of all (finite) sums of the form $v_{i_1}+\ldots+v_{i_n}$ where $v_{i_k}\in V_{i_k}$ and $V_{i_k}\subseteq X$.