Taylor series

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)}
$



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