(Taylor decomposition in Peano
form) If a function
is
times differentiable at
then

Proposition

(Taylor decomposition in Schlomilch, Lagrange and Cauchy forms) If
a function
is
times differentiable in
,
then
where
admits any of the following representations:

(a)

(b)

(c)

for
and
.

Proposition

(Integral form of Taylor
decomposition) If a function
has
derivatives in finite interval between
and
,
and
is in
in such interval
then