Prove that if the function is infinitely differentiable on an interval containing , then for any and any positive integer we may expand in the form
where the remainder term should be specified explicitly in terms of .
Let be a nonzero polynomial in , and let be the real function defined by
Show that is differentiable everywhere and that
where . Deduce that is infinitely differentiable, but that there exist arbitrarily small values of for which the remainder term in the Taylor expansion of about 0 does not tend to 0 as .