Expand the function into Maclaurin series on the whole real line .

So here is what I have done so far:

From what I have been able to figure out, though I am not sure this is correct, the Maclaurin series is defined as

.

I solved for .

Similarly, , giving .

Once again, , giving .

Is the right direction? Should I be finding some pattern here? What is the final answer supposed to look like?