Let $R$ denotes the ring of infinite sequences over rational, written by vectors $\bar{s}=(a_1,a_2,\cdots).$
In the power series ring $R[[x]]$, a general element is of type $\bar{s}_0 + \bar{s}_2x + \bar{s}_2x^2+\cdots $, so it is a power series in $x$ with coefficients the vectors/sequences over rationals.
Let $I$ denote the ideal of $R$ generated by $$\bar{e}_1=(1,0,\cdots), e_2=(0,1,\cdots),\cdots. $$ So $I$ contains sequences which are zero after finite steps.
Let $IR[[x]]$ denote the ideal in $R[[x]]$ generated by set $I$.
Q. I do not understand why $I[[x]]$ is bigger than $IR[[x]]$?
(I confused with elements of finite support in both ideals. )