# When can a function be taken inside a sum?

Consider a convergent sum of the form $$A(x) = \sum_{n=0}^\infty c\, a_n(x)\, x^n.$$ This sum is also bounded as $$x \rightarrow \infty$$, for example, it is a Taylor series for $$A(x)$$ at $$0$$.

I am interested in the product of $$A(x)$$ with $$B(x)$$, where $$B(x)$$ is bounded $$\forall\, x$$, $$B(x)A(x) = B(x)\sum_{n=0}^\infty c_n\, a_n(x)\, x^n.$$ I want to take $$B(x)$$ into the sum to get $$B(x)A(x) = \sum_{n=0}^\infty c_n\, B(x)\, a_n(x)\, x^n.$$ Under what conditions is this sum also guaranteed to converge, particularly as $$x \rightarrow \infty$$?

Note that I know the product $$B(x)\, a_n(x)\, x^n$$ is bounded $$\forall\,x$$.

I found this question and this question which seem similar. Apologies if this is a duplicate, but I'm not sure if I can apply their answers to my situation. Can I simply use the fact that $$c \sum_n a_n = \sum_n c\, a_n$$ for bounded $$c$$? Or are there any special considerations, due to the fact that $$B$$ depends on $$x$$ which also occurs in the sum?

Also, does anything change if $$B(x)$$ is complex?

For any $$x$$ you can always take $$B(x)$$ inside the sum. This does follow from the fact that $$c \sum a_n =\sum ca_n$$. It is not clear what you mean by 'particularly as $$x \to \infty$$'