You sure you are doing your derivatives right in the MacLaurin Polynomial?

I think you forgot a few things like chain rule and I am not sure what else went wrong when you did it.

So i get like

I did it kinda fast, so you should look over it to be sure, but it looks like this works out. I think the pattern really gets going here starting at the second term of the expansion. There might be a slicker way of writing it, but I think this will do.