Direct Limit of Power Series

Question

Consider the following directed system of power series $$R[[X_1]]\hookrightarrow R[[X_1,X_2]]\hookrightarrow\cdots\hookrightarrow\varinjlim_iR[[X_1,\ldots, X_i]]=\bigcup_i R[[X_1,\ldots, X_i]] $$ Question 1 Is $R[[X_1,X_2,\ldots]]=\bigcup_i R[[X_1,\ldots, X_i]]$? This is true for polynomial rings.
Question 2 If above not true, how can we express $R[[X_1,X_2,\ldots]]$ as a direct limit of Notherian rings?
Edit: Let $f=X_1+X_2+\ldots+X_k+\ldots$, then $f\in R[[X_1,X_2,\ldots]]$, but $f\notin R[[X_1,\ldots, X_i]]$ for any $i$, thus $f\notin\bigcup_i R[[X_1,\ldots, X_i]] $. The universal property of direct limit now gives $\bigcup_i R[[X_1,\ldots, X_i]] \subsetneq R[[X_1,X_2,\ldots]]$. The example given by @logarithm can also be used. But, Question 2 is still open.