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

Tn(x,a) = f(a) + (x−a) + (x−a)2 + … + (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

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

f(x) =  f(a)+(x−a)+(x−a)2+ … +(x−a)n+R n(x)

(2)

with

where ξ 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

∑ n=0∞(x−a)n = f(a) + (x−a) + (x−a)2 + …

(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

f(x) = f(a) + (x−a) + (x−a)2 + …

(4)

The most known Taylor series is perhaps

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: