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Taylor's theorem (Theorem)

Taylor's Theorem

Let $ f$ be a function which is defined on the interval $ (a,b)$ and suppose the $ n$th derivative $ f^{(n)}$ exists on $ (a,b)$. Then for all $ x$ and $ x_0$ in $ (a,b)$,

$\displaystyle R_n(x) = \frac{f^{(n)}(y)}{n!}(x-x_0)^n $

with $ y$ strictly between $ x$ and $ x_0$ ($ y$ depends on the choice of $ x$). $ R_n(x)$ is the $ n$th remainder of the Taylor series for $ f(x)$.



"Taylor's theorem" is owned by Andrea Ambrosio. [ full author list (2) | owner history (1) ]
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See Also: Taylor series


Attachments:
proof of Taylor's Theorem (Proof) by rmilson
example of use of Taylor's theorem (Example) by alozano
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Cross-references: Taylor series, remainder, strictly, derivative, interval, function
There are 7 references to this entry.

This is version 7 of Taylor's theorem, born on 2001-11-08, modified 2006-06-23.
Object id is 706, canonical name is TaylorsTheorem.
Accessed 12222 times total.

Classification:
AMS MSC: 41A58 (Approximations and expansions :: Series expansions )
Pending Errata and Addenda
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