isomorphism between $k[[x]]$ into $\varprojlim_n k[x]/(x^n)$

Question

i want to find isomorphism between $k[[x]]$ and $\varprojlim_n k[x]/(x^n)$ but I cant.please help me to find this.

Answer 1

Let $A$ be your projective limit. Prove that every element in $A$ can be uniquely written as a convergent power series of powers of $x$.

Let $a\in A$, this means that, $$ a = \left( a_0, a_1, a_2, ... \right) $$ Where $a_j \in k[x]/(x^j)$ i.e. a polynomial of degree $j$.

Furthermore, the projection map $k[x]/(x^{j+1}) \to k[x]/(x^j)$ has to send $a_{j+1}\mapsto a_j$. So if you choose the $(j+1)$-st term of $a_{j+1}$ it will agree with the $j$-th term of $a_j$. Call these term $b_j$. Now if you form the infinite series $\sum_{j=0}^{\infty} b_j$ it will converge, in the $(x)$-adic topology $A$, to $a$.

Thus, define the map $A\to k[[x]]$ in this manner.