(calculus) A power series representation of given infinitely differentiable function \textstyle f whose terms are calculated from the function's arbitrary order derivatives at given reference point \textstyle a; the series \textstyle f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f(a)}{2!} (x-a)^2+\frac{f(a)}{3!}(x-a)^3+ \cdots = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^{n}.