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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