If
is
the degree
Taylor polynomial at zero and
is
the error with which
approximates
, then the Taylor
remainder theorem gives

on any given interval
which contains zero. The
constant
depends on the interval and satisfies
. To estimate the
error
we will need to choose the interval
so that both and
belong to it. The interval of
minimal length containing both and
is
. The
calculation of the derivatives of
gives
and so we have
since the function
is monotonically increasing on the
interval
.
Therefore
on the interval
and so
. By plugging in
the first few values of
we find that
is the first possible
value of
for which
.