Suppose $f(x)$ is a smooth function and $g_n(x)$ is a sequence of functions such that $$ \sum_{n=1}^\infty f^{(n)}(x) g_n(x) $$ converges for $x \in [a,b]$, where $f^{(n)}(x)$ denotes the $n$-th derivative of $f(x)$. Can we make any conclusions about the convergence of the "shifted" series
\begin{equation} \label{1} \sum_{n=1}^\infty f^{(n+1)}(x) g_n(x), \end{equation} or at least determine if $\displaystyle \lim_{n\rightarrow \infty} f^{(n+1)}(x) g_n(x) = 0, \ \forall \, x \in [a,b]$?
If not, what additional conditions would I need to ensure either the convergence of the shifted series or of the shifted limit? Thank you.