I'm confused between some formulae so I'm going to give some examples and you can let me know if what I'm writing is correct.

Find the Taylor series for....

EXAMPLE 1:

around

Then,

provided that or,

Does this look correct?

EDIT: Now if I do the same question in another fashion...

provided that or,

They aren't equal, in this one I have a kicking around in my sum.

What am I doing wrong here?