Definition:Taylor Series
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Definition
Let $f$ be a real function which is smooth on the open interval $\openint a b$.
Let $\xi \in \openint a b$.
Then the Taylor series expansion of $f$ about the point $\xi$ is:
- $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
It is not necessarily the case that this power series is convergent with sum $\map f x$.
Remainder
Consider the Taylor series expansion $\map T {\map f \xi}$ of $f$ about the point $\xi$:
- $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
Let $\map {T_n} {\map f \xi}$ be the Taylor polynomial:
- $\ds \sum_{n \mathop = 0}^n \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$
for some $n \in \N$.
The difference:
- $\ds \map {R_n} x = \map f x - \map {T_n} {\map f \xi} = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$
is known as the remainder of $\map T {\map f \xi}$ at $x$.
Also see
Source of Name
This entry was named for Brook Taylor.
Historical Note
The first proof for the convergence of a Taylor series was provided by Augustin Louis Cauchy.
He used the Cauchy Form of the remainder, showing that it converges to zero.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Taylor Series for Functions of One Variable
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 15.4$
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3.2$: Power series: $(1.42)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Taylor's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Taylor's theorem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Taylor polynomial, Taylor series (expansion)