Taylor series
TaylorSeries
2009-05-18 13:58:16
2009-05-19 14:36:53
Topic
introduced X, Y, Z
Taylor polynomial
Taylor formula
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Any power series represents on its convergence domain a function.\, One may set a converse task:\, If there is given a function $f(x)$, on which conditions one can represent it as a power series; how one can find the coefficients of the series?\, Then one comes to Taylor polynomials, Taylor formula and Taylor series.
\textbf{Definition.}\, The \emph{Taylor polynomial} of degree $n$ of the function $f(x)$ in the point\, $x = a$\, means the polynomial $T_n(x,a)$ of degree at most $n$, which has in the point the value $f(a)$ and for which the derivatives $T_n^{(j)}(x,a)$ up to the order $n$ have the values $f^{(j)}(a)$.
It is easily found that the Taylor polynomial in question is uniquely
\begin{align}
T_n(x,a) \;=\; f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+
\ldots+\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n
\end{align}
When a given function $f(x)$ is replaced by its Taylor polynomial $T_n(x,a)$, it's important to examine, how accurately the polynomial approximates the function, in other words one has to examine the difference
$$f(x)\!-\!T_n(x,a) \;:=\; R_n(x).$$
Then one is led to the\\
\textbf{Taylor formula.}\, If $f(x)$ has in a neighbourhood of the point\,
$x = a$\, the continuous derivatives up to the order $n\!+\!1$, then it can be represented in the form
\begin{align}
f(x) \,=\ f(a)\!+\!\frac{f'(a)}{1!}(x\!-\!a)\!+\!\frac{f''(a)}{2!}(x\!-\!a)^2\!+
\ldots+\!\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n\!+\!R_n(x)
\end{align}
with
$$R_n(x) \;=\; \frac{f^{(n+1)}(\xi)}{(n\!+\!1)!}(x\!-\!a)^{n+1}$$
where $\xi$ lies between $a$ and $x$.\\
If the function $f(x)$ has in a neighbourhood of the point\, $x = a$\, the derivatives of all orders, then one can let $n$ tend to infinity in the Taylor formula (2).\, One obtains the so-called \emph{Taylor series}
\begin{align}
\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n \;=\;
f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+\ldots
\end{align}
\textbf{Theorem.}\, A necessary and sufficient condition for that the Taylor series (3) converges and that its sum represents the function $f(x)$ at certain values of $x$ is that the limit of $R_n(x)$ is 0 as $n$ tends to infinity.\, For these values of $x$ on may write
\begin{align}
f(x) \;=\;
f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+\ldots
\end{align}
The most known Taylor series is perhaps
$$e^x \;=\; 1+\frac{x}{1!}+\frac{x^2}{2!}+\ldots$$
which is valid for all real (and complex) values of $x$.\\
There are analogical generalisations of Taylor theorem and series for functions of several real variables; then the existence of the partial derivatives is needed.\, For example for the function $f(x,y,z)$ the Taylor series looks as follows:
$$
f(x,y,z) \;=\;
f(a,b,c)+\sum_{n=1}^{\infty}
\left[\frac{1}{n!}\!\left(
(x\!-\!a)\frac{\partial}{\partial x}+
(y\!-\!b)\frac{\partial}{\partial y}+
(z\!-\!c)\frac{\partial}{\partial z}\right)^n\!f\right]_{(x,y,z)=(a,b,c)}
$$