This tag is for questions relating to "formal power series" which can be considered either as an extension of the polynomial to a possibly infinite number of terms or as a power series in which the variable is not assigned any value.

A **formal power series**, sometimes simply called a "**formal series**" is, informally, an expression of the form $$a(x) = a_0 +a_1x+a_2x^2 +a_3x^3 +\cdots=\sum_{i=0}^{\infty}a_i~x^i~,$$where the $a_i$ are numbers, but it is understood that no value is assigned to $x$. Unlike usual power series, convergence is not usually a concern.

Addition (termwise) and multiplication (Cauchy product) operations are defined, allowing formal power series to be studied in the context of ring theory. The set of all formal power series in $X$ with coefficients in a commutative ring $R$ forms another ring called the **ring of formal power series** in the variable $X$ over $R$, commonly written $R[[X]]$. Formal power series can be created from Taylor polynomials using formal moduli.

A related concept is a formal Laurent series, where the sum can be allowed to take finitely many negative values. The ring of formal Laurent series in $X$ over $R$ is denoted $R((X))$.

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