Hi Everyone! Today we are asked to find a Taylor Series for $\displaystyle \sqrt{x}$ at about its center, a=4. I've been working on this one problem for hours now, and I feel so dumb not being able to find a Taylor series for it. So far I've been working on $\displaystyle f^n(x)$formulas. So far, the nth derivatives goes like this:$\displaystyle \frac{1}{4},\frac{-1}{32},\frac{3}{256},\frac{-15}{2048},\frac{105}{16384}...$. The easy part is that the denominator is a base 2 power function and can be given by $\displaystyle 2^{3n-1}$. The alternating signs can be given by $\displaystyle (-1)^{n+1}$. But the numerator is the real problem. It can be given by 1,1,1*3,1*3*5,1*3*5*7... which is somewhat equal to simply $\displaystyle \frac{(2n-1)!}{2^{n-1}(n-1)!}$. But the problem is, instead of the numerator going: 1,3 ,15,105... it instead goes: 1,1,3,15,105...

That first numerator is giving me problems. Anyone got any ideas on how I should proceed with this or at least, how to give find the Taylor series for this function? Thanks everyone in advance.