I was reading Amann and Escher's Analysis I. I encountered this notation of formal power series on page 72
and got confused.
Could someone please tell me what do the subscript(in this case "n") and the superscript (in this case "m") mean respectively?
(I was confused because when n = 2, $X_2 = 0$, then $X_2^2 = 0$, but according to the second function, it should equal 1, what's wrong with my reasoning?)
P.S. The definition of the R[[N]] in the question is defined as following in the book
$R[[X]]$ is introduced in a very formal and abstract but absolutely correct way. According to this definition an element of $R[[X]]$ is nothing else than an infinite sequence $\mathbf r= (r_0,r_1,r_2,\ldots)$ with $r_i \in R$.
$X$ is defined as the sequence $(0,1,0, 0,\ldots)$. Its $m$-th power $X^m$ in the ring $R[[X]]$ is given as $(0,\ldots,0,1,0,,0,\ldots)$ where $1$ occurs at position $m$. Then $X^0 = (1,0,0,\ldots)$, $X^1 = X$, $X^2 = (0,0,1,0,\ldots)$ etc.
An arbitrary element of $R[[X]]$ is then usually written as $$\mathbf r = (r_0,r_1,r_2,\ldots) = \sum_{m=0}^\infty r_mX^m .$$ Note, however, that infinite sums in general do not exist in $R[[X]]$. An infinite sum $$\sum_{m=0}^\infty\mathbf r_m$$ is well-defined precisely if for each $i$ the number of $\mathbf r_m$ with $i$-th coordinate $(\mathbf r_m)_i \ne 0$ is finite.