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Taylor series (Topic)

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.

Definition.  The 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

$\displaystyle T_n(x,a) \;=\; f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+ \ldots+\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n$ (1)

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

$\displaystyle f(x)\!-\!T_n(x,a) \;:=\; R_n(x).$
Then one is led to the

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

$\displaystyle f(x) \,=\ f(a)\!+\!\frac{f'(a)}{1!}(x\!-\!a)\!+\!\frac{f''(a)}{2!}(x\!-\!a)^2\!+ \ldots+\!\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n\!+\!R_n(x)$ (2)
with

$\displaystyle R_n(x) \;=\; \frac{f^{(n+1)}(\xi)}{(n\!+\!1)!}(x\!-\!a)^{n+1}$
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 Taylor series

$\displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x\!-\!a)^n \;=\; f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+\ldots$ (3)

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

$\displaystyle f(x) \;=\; f(a)+\frac{f'(a)}{1!}(x\!-\!a)+\frac{f''(a)}{2!}(x\!-\!a)^2+\ldots$ (4)

The most known Taylor series is perhaps

$\displaystyle e^x \;=\; 1+\frac{x}{1!}+\frac{x^2}{2!}+\ldots$
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:

$\displaystyle f(X,Y,Z) \;=\; f(a,b,c)+\sum_{n=1}^{\infty} \left[\frac{1}{n!}\!\... ... y}+ (Z\!-\!c)\frac{\partial}{\partial z}\right)^n\!f\right]_{(x,y,z)=(a,b,c)} $



"Taylor series" is owned by pahio.

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Also defines:  Taylor polynomial, Taylor formula

Cross-references: theorem, function, domain, power
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This is version 6 of Taylor series, born on 2009-05-18, modified 2009-05-19.
Object id is 762, canonical name is TaylorSeries.
Accessed 778 times total.

Classification:
Physics Classification: 02.30.-f (Function theory, analysis)
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