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