Noun
Taylor series (plural Taylor series)
(calculus) A power series representation of given infinitely differentiable function
f
{\displaystyle \textstyle f}
whose terms are calculated from the function's arbitrary order derivatives at given reference point
a
{\displaystyle \textstyle a}
; the series
f
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a
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+
f
′
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1
!
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f
″
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2
!
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2
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f
‴
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3
!
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3
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⋯
=
∑
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∞
f
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n
{\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}}
.
Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. Source: Internet
Applying the multi-index notation the Taylor series for several variables becomes : which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case. Source: Internet
As a result, the radius of convergence of a Taylor series can be zero. Source: Internet
A function that is equal to its Taylor series in an open interval (or a disc in the complex plane ) is known as an analytic function in that interval. Source: Internet
And even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f(x). Source: Internet
A function may not be equal to its Taylor series, even if its Taylor series converges at every point. Source: Internet