The Taylor polynomial of order 
for a function 
about 
is the polynomial
 |
where 
is a nonnegative integer and the derivatives through
order 
exist at 
.
Its polynomial degree is at most 
. It agrees with 
and its first 
derivatives at 
.
When 
has a Taylor series about 
, its Taylor polynomial of order 
consists of the first 
terms. Unlike the infinite Taylor
series, this polynomial requires only the first 
derivatives at 
and does not require convergence
of the series to 
.
The difference 
is the Taylor remainder. For example, the Taylor
polynomial of order 2 for 
about 0 is 
.
