Taylor Series

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A Taylor Series represents a expansion of a function around a given point. Uses include trigonometry and the natural exponent.

For an example, we will consider the natural exponent, e around 0:

$e = 1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!}$

Expansion of e using Taylor Series may be useful in derivations or where $e x$ features small values of x.

A stimulating discussion of Taylor Series may be found in Comtet's "Calcul pratique des coefficients de Taylor d'une fonction algébrique" (Enseign. Math. 10, 267-270, 1964) as well as Whittaker and Watson's landmark treatise, "Forms of the Remainder in Taylor's Series." found in A Course in Modern Analysis, 4th ed.

Source: Wolfram MathWorld: http://mathworld.wolfram.com/TaylorSeries.html.

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