Any power
series represents on its convergence domain a function. One may set a converse task: If there is given a function
, 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
of the function
in the point
means the polynomial
of degree at most
, which has in the point the value
and for which the derivatives
up to the order
have the values
.
It is easily found that the Taylor polynomial in question is uniquely
![]() |
(1) |
When a given function
is replaced by its Taylor polynomial
, 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
has in a neighbourhood of the point
the continuous derivatives up to the order
, then it can be represented in the form
![]() |
(2) |
where
If the function
has in a neighbourhood of the point
the derivatives of all orders, then one can let
tend to infinity in the Taylor formula (2). One obtains the so-called Taylor series
![]() |
(3) |
theorem. A necessary and sufficient condition for that the Taylor series (3) converges and that its sum represents the function
at certain values of
is that the limit of
is 0 as
tends to infinity. For these values of
on may write
![]() |
(4) |
The most known Taylor series is perhaps
which is valid for all real (and complex) values of
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
the Taylor series looks as follows:
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