Beyond direct application of the Taylor series formula (summing derivatives at a point divided by factorials, times powers of x), coefficients of a power series can often be found more efficiently by exploiting algebraic relationships to other functions whose series are already known: using symmetry properties (e.g., an even function's series has only even-power terms) to eliminate terms, or using the fact that power series multiply like polynomials so that a known functional identity between two functions translates into a system of equations relating their series' coefficients. This is a topic in single-variable calculus within the theory of Taylor and power series, connecting series computation to function symmetry, algebraic identities between functions, and formal power series arithmetic (including power series long division).
Beyond direct application of the Taylor series formula (summing derivatives at a point divided by factorials, times powers of x), coefficients of a power series can often be found more efficiently by…