Taylor series

See also: Taylorseries

English

Alternative forms

• Taylor's series

Etymology

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

Noun

Taylor series (plural Taylor series)

1. (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 $\textstyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}$.
   • 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^[1], 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.

Hyponyms

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

Translations

power series of a function calculated from derivatives at a reference point

• Chinese:

  Mandarin: 泰勒展開式／泰勒展开式 (Tàilè zhǎnkāishì)

• Czech: Taylorova řada f
• Danish: Taylorrække c
• German: Taylorreihe f
• Hungarian: Taylor-sor (hu)
• Italian: serie di Taylor (it) f
• Romanian: serie Taylor f
• Russian: ряд Те́йлора m (rjad Tɛ́jlora)