How can I show that
is infinitely differentiable?
The first derivative is
The second derivative is
The third derivative is
The fourth derivative is
So
(You might want to prove this using induction, though it will likely be ugly.)
Next you'd have to do another induction proof. Like assume that and then prove that:
Since you assumed , you need to prove that
.
There could be a cleaner way to do this using some theorems or something, but this should work.