Now that CaptainBlack and Jhevon have helped me out with Latex notation I can ask my question.

I was working on taylor polynomials and I ran into a little snag which has me stumped.

For the function

we get the taylor series:

So for

we get

Yet if we construct the taylor polynomial for

from scratch:

for x= 0

in fact we dont see a non zero constant term until the 12th order differential:

Which looks quite crazy I've just realised. Some of the terms are

.

So if we get this, then the numerator on the second term of the taylor polynomial should be massive. So why does letting

work? I understand the logic behind it I suppose, as the taylor polynomial work for any value of

and

is just a finite number (for any finite

and for this

function, I know it doesn't hold for all functions). But why does the method of constructing the taylor polynomial from the formula:

fail?