Formal power series in the context of Convergent series

In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).

A formal power series is a special kind of formal series, of the form $${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots ,}$$where the ${\displaystyle a_{n},}$ called coefficients, are numbers or, more generally, elements of some ring, and the ${\displaystyle x^{n}}$ are formal powers of the symbol ${\displaystyle x}$ that is called an indeterminate or, commonly, a variable. Hence, formal power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a formal power series may not represent a function of its variables. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.