Proof of smooth functions

[hobz]

Suppose I have f(x)=e^x, how can it be proven that it is smooth?

[the tree]

A function is smooth if it has a derivative of all orders, so you simply have to find its derivatives for all orders. You should be able to prove that by induction.

[hobz]

I see! So the implicit requirement for a function to be smooth in order to be differentiable, can be used "in reverse" to find out if a function is smooth.

 

This also implies that all smooth functions can be represented as a Taylor series?

[the tree]

Yes they can.

[hobz]

Thanks!