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Revision difference : Taylor's theorem
Version 5 Version 4
\section{Taylor's Theorem} \section{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 $x$ an $x_0$ in $(a,b)$, Let $f$ be a function which is defined on the interval $(a,b)$, with $a<0<b$, and suppose the $n$th derivative $f^{(n)}$ exists on $(a,b)$. Then for all nonzero $x$ in $(a,b)$,
$$ R_n(x) = \frac{f^{(n)}(y)}{n!}(x-x_0)^n $$ $$ R_n(x) = \frac{f^{(n)}(y)}{n!}x^n $$
with $y$ strictly between $x$ and $x_0$ ($y$ depends on the choice of $x$ and $x_0$). $R_n(x)$ is the $n$th remainder of the Taylor series for $f(x)$. with $y$ strictly between 0 and $x$ ($y$ depends on the choice of $x$). $R_n(x)$ is the $n$th remainder of the Taylor series for $f(x)$.