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Taylor’s Expansion

  • Richard Courant
  • Fritz John

Abstract

It was a great triumph in the early years of Calculus when Newton and others discovered that many known functions could be expressed as “polynomials of infinite order” or “power series,” with coefficients formed by elegant transparent laws. The geometrical series for 1/(1 − x) or 1/(1 + x2)
$$\frac{1}{{1\; - \;x}} = 1 + x + {x^2} \cdots + {x^n} + \cdots $$
(1)
$$\frac{1}{{1 + {x^2}}} = 1 - {x^2} + {x^4} - {x^6} + \cdots + {( - 1)^n}{x^{2n}} + \cdots $$
(1a)
valid for the open interval |x| < 1, are prototypes (see Chapter 1, p. 67).

Keywords

Taylor Series Interpolation Formula Continuous Derivative Taylor Formula Taylor Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1989

Authors and Affiliations

  • Richard Courant
    • 1
  • Fritz John
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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