PlanetPhysics/Taylor Series

Any [[../Power/|power]] series represents on its convergence [[../Bijective/|domain]] a [[../Bijective/|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

$\begin{array}{c}{T}_n(x,a)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\dots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n\end{array}$

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\\

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

$\begin{array}{c}f(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\dots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n+R_n(x)\end{array}$

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 Taylor series

$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\dots \end{array}$

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.}\, 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

$\begin{array}{c}f(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\dots \end{array}$

The most known Taylor series is perhaps $e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\dots$ 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: $f(X,Y,Z)=f(a,b,c)+\sum _{n=1}^{\mathrm{\infty }}{\left[\frac{1}{n!}{((X-a)\frac{\partial }{\partial x}+(Y-b)\frac{\partial }{\partial y}+(Z-c)\frac{\partial }{\partial z})}^{n}f\right]}_{(x,y,z)=(a,b,c)}$