I am attempting to justify the expansion
$ \sqrt{1+x}= 1 + \frac{x}{2} + \sum_{n=2}^{\infty}{(-1)^n \frac{1}{2n}\frac{(1-\frac{1}{2}) \cdots ((n-1)-\frac{1}{2})}{(n-1)!}x^n} $
for -1
I've got the expansion, but I cannot prove that the error term tends to zero.
$ E_n = \frac{(-1)^{n-1}(2n-3)(2n-1) \cdots (1)}{2^n n!}(1+\theta x)^{-\frac{2n-1}{2}}x^n $
where $\theta \in (0,1)$
The question suggests using the Constancy Lemma (if the differential is zero, the function is constant), but I can't make that work either. Any help gratefully appreciated.
(While technically this is homework I'm the tutor so I allowed to cheat! Also it is extra embarrassing that I cannot do it)