Taylor series

• Taylor's series

Named after English mathematician Brook Taylor, who formally introduced the series in 1715. The concept was formulated by Scottish mathematician James Gregory.

Taylor series (plural Taylor series)

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 ${\displaystyle \textstyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}$.
   • 1978, [McGraw-Hill], Carl M. Bender, Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, published 1999, page 324:

     A series solution about an ordinary point of a differential equation is always a Taylor series having a nonvanishing radius of convergence. A series solution about a singular point does not have this form (except in rare cases). Instead, it may be either a convergent series not in Taylor series form (such as a Frobenius series) or it may be a divergent series.

   • 1980, Suhas Patankar, Numerical Heat Transfer and Fluid Flow‎, Taylor & Francis (CRC Press), page 28:

     The usual procedure for deriving finite-difference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series.

   • 1998, Kenneth L. Judd, Numerical Methods in Economics, The MIT Press, page 197:

     This function has its only singularity at x = 0, implying that the radius of convergence for the Taylor series around x = 1 is only unity.

• (power series of a function calculated from derivatives at a reference point): Maclaurin series