If it were a problem of the form 1/(1-x) I'd know what to do but it isn't. Any ideas?
This is very peculiar- you use the terms "Taylor Polynomial" and "Taylor Series" but your questions imply that you don't know the definition or formula!
The "third degree Taylor Polynomial" for function f(x) about x= a is
.
Here, . .
so .
Can you get f''(4) and f'''(4)?