Let $ p$ be a prime number. Consider ring of formal power series in $ p$ with coefficients from $ \mathbb{Z}/p\mathbb{Z}$ i.e., the set
$$F=\{a_0+a_1p+a_2p^2+\cdots : a_i\in \mathbb{Z}/p\mathbb{Z}\}.$$
Addition and multiplication is defined as addition and multiplication of formal power series.

For the same $p$ as above, we define what is called completion of $\mathbb{Z}$ with respect to $p$.
What ever it may be, as a set it is
$$C=\{(x_1,x_2,\cdots): x_n\in \mathbb{Z}/p^n\mathbb{Z}, x_{n+1}-x_n\in p^n\mathbb{Z}\}.$$
Addition and multiplication is defined componentwise.

We want to see that these two gives isomorphic rings. For that, we at least need to have some bijective correspondence between $F$ and $G$.

Let $(x_1,x_2,\cdots)\in C$. Then, we want to get an element of the form $a_0+a_1p+\cdots$ with $a_i\in \mathbb{Z}/p\mathbb{Z}$.

We want $a_0\in \mathbb{Z}/p\mathbb{Z}$. We have $x_1\in \mathbb{Z}/p\mathbb{Z}$. Set $a_0=x_1$.

We want $a_1\in \mathbb{Z}/p\mathbb{Z}$. We see that $x_2\in \mathbb{Z}/p^2\mathbb{Z}$ such that $x_2-x_1\in p\mathbb{Z}$ i.e., $p$ divides $x_2-x_1$. Let $x_2-x_1=pt$ then $t<p$ as $x_2,x_1<p^2$.

Set $a_1=t$. See that
$x_2=a_0+a_1p$.

With this we can guess what should the other $a_n$ have to be. We define $a_n=\dfrac{x_{n+1}-x_{n}}{p^n}\in \mathbb{Z}/p\mathbb{Z}$.

We also have $x_n=a_0+a_1p+a_2p^2+\cdots+a_{n-1}p^{n-1}$.

So, given $(x_1,x_2,\cdots)$ we have an element $a_0+a_1p+a_2p^2+\cdots$.

Let $a_0+a_1p+a_2p^2+\cdots$. We have to assign $(x_1,x_2,\cdots)$ with properties as above. By above observation, it is natural to assign $x_n=a_0+a_1p+a_2p^2+\cdots$

It is clearly bijective and it is upto you to check the isomorphism as rings.