Infinite sum in power series ring

Let $$R$$ be a commutative ring with $$1$$, $$R[[x]]$$ be the power series ring over $$R$$ and $$A$$ be an (prime) ideal of $$R[[x]]$$ with $$xnotin A$$ and $${f_i}_{i=1}^infty$$ be a sequence of element of $$A$$. Now I have two questions:

1. Is $$fmathrel{:=}f_1+xf_2+x^2f_3+dotsb+x^nf_{n+1}+dotsb$$ a well defined element of $$R[[x]]$$?
(Since we can find the coefficient of $$x^n$$ in $$f$$ for each $$n$$, it seems that $$f$$ is well defined.)

2. If (1) is true is $$fin A$$? (If (1) is true and (2) is not true, under what conditions is (2) true?)

MathOverflow Asked on November 9, 2021

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