Formal power series

A formal power series is an infinite sum that looks like a polynomial, but with infinitely many terms. It is written as $a_0 + a_1x + a_2x^2 + a_3x^3 + ext{...}$, where the $a_i$ are coefficients from some number system (like the real numbers or integers). Unlike regular power series in calculus, we do not worry about whether the sum 'converges' for any value of $x$—we treat $x$ as a symbol and focus on algebraic manipulation.

Important properties

• Formal power series can be added and multiplied using the same rules as polynomials.

• There is no need to substitute a value for $x$ or worry about convergence.

• The set of all formal power series with coefficients in a ring forms a ring itself, called the ring of formal power series.