I have the following problem, to show that when $z^2-1$ is not a positive number, that the following holds:
$$\frac{1}{\sqrt{1-z^2}}=\sum_{k=0}^{n}\frac{(2k-1)!!}{(2k)!!}z^{2k}+\frac{(2n+1)!!}{(2n)!!}\frac{1}{\sqrt{1-z^2}}\int^z_0\frac{t^{2n+1}}{\sqrt{1-t^2}}dt$$
This result is apparently due to Jacobi and Scheibner. It is easy to find the derivatives of the function and find the Taylor series and when it converges, but I don't quite see how the remainder term was calculated, and it's not something I'm used to doing so I'm not sure how to go about putting it in this form.
Jacobi and Scheibner have many results that take a very similar form to this and I would like to understand how they got there.
An answer or even a helpful hint would be appreciated. Thanks.