Hey all I know this question has been asked before, but the other thread has not answered the question I am looking for. I think I found a series representation of $f(x) = \sqrt{x}$, but it uses the double factorial $$1 + \frac{1}{2}(x-1)+\sum_{n=2}^{\infty}(-1)^{n-1}\frac{(2n-3)!!(x-1)^n}{2^nn!}$$
I am fairly sure this representation is correct, but my professor has never introduced us to double factorials so I have a feeling this is not the way he wants us to approach this problem. I have been introduced to binomial series and I feel that might be a better representation, but I am still unsure how to write it as a binomial series, any pointers? Thanks!
Let $x=1+y$ and consider the expansion of $\sqrt{1+y}$ around $y=0$.
Then using the binomila theorem, you just have $$\sqrt{1+y}=\sum_{n=0}^\infty \binom{\frac{1}{2}}{n} y^n$$ Back to $x$ $$\sqrt{x}=\sum_{n=0}^\infty \binom{\frac{1}{2}}{n} (x-1)^n$$