A power series
where the numerical function
is defined in some neighbourhood of the point
and has at this point derivatives of all orders. The
partial sums of a Taylor series are Taylor polynomials (cf.
Taylor polynomial).
If
is a complex number and the function
is defined in some neighbourhood of
in the field of complex numbers and is differentiable at
,
then there exists a neighbourhood of
in which
is the sum of its Taylor series
(1)
(see
Power series).
If, however,
is a real number and the function
is defined in some neighbourhood of
in the field of real numbers and has at
derivatives of all orders, then there may be no neighbourhood of
in which
is the sum of its Taylor series. For instance, the function
is differentiable infinitely many times on the entire real
axis, is equal to 0 only at the point
,
but all coefficients of its Taylor series at this point are equal to 0.
If a function is the sum in some neighbourhood of a
given point of a power series with centre at that point,
then such a series is unique and is the Taylor series of
this function at the given point. However, one and the same power
series can be the Taylor series of different real functions. Indeed, the power
series all coefficients of which are equal to 0 is the Taylor series
both of the function identically equal to 0 on the entire
real axis and of the function
(2)
at the point
.
A sufficient condition for the convergence of the
Taylor series
(1)
to the real-valued function
on an interval
is the existence of a common bound for all its derivatives in this interval.
The Taylor series can be generalized to the case of mappings of subsets
of linear normed spaces into similar spaces, and in particular to numerical
functions of several variables and to functions of a matrix argument.
The series
(1)
was published by
B. Taylor
in
1715,
a series reducible
to the series
(1)
by a simple transformation
was published by
Johann I. Bernoulli
in
1694.