I made a mistake, I've edited my post. However I still don't reach the Taylor series you provided. I don't see why the series I found is not a Taylor series.

I've used the fact that

.

I've found out that

.

Now you make me doubt the formula holds for the function

due to the singularity at

.

Edit : I understand the equality you gave, but I don't know how to reach it. I mean I don't know how to reach that the Taylor Series is

Edit 2 : Sorry, I just found my error. I forgot to evaluate

in

...

I think I'll do it. Thanks a lot!