Calc 2 - Taylor Expansion Series of x^(1/2)

[omg4cards]

Homework Statement

f(x) = $$\sqrt{x}$$, a = 4

Homework Equations

f(x) = $$\sum$$f$$^{n}$$(a)/n! (x-a)$$^{n}$$

The Attempt at a Solution

f(x) = x$$^{1/2}$$
f$$^{'}$$(x) = $$\frac{1}{2}$$x$$^{1/2}$$
f$$^{2}$$(x) = -$$\frac{1}{2}$$*$$\frac{1}{2}$$x$$^{-3/2}$$
f$$^{3}$$(x) = $$\frac{1}{2}$$*$$\frac{1}{2}$$*$$\frac{3}{2}$$x$$^{-5/2}$$
f$$^{4}$$(x) = -$$\frac{1}{2}$$*$$\frac{1}{2}$$*$$\frac{3}{2}$$*$$\frac{5}{2}$$x$$^{-7/2}$$

f$$^{n}$$(x) = (-1)$$^{n+1}$$*$$\frac{1}{2}$$$$^{n}$$*x$$^{-[(2n-1)/2]}$$*?


The problem I am having here is with identifying the pattern. I am able to describe everything except the numbers in the numerator(1, 1*1, 1*1*3, 1*1*3*5...). Any help is greatly appreciated!

 

[omg4cards]

I have no idea why it looks like that...

 

[mg0stisha]

shouldn't the number's in the numerator be (1*-1*-3*-5*...)?

 

[omg4cards]

I left out the signs to simplify and becaus I already identified the pattern w/ (-1)^(n+1)... so if i kept the signs in, the #'s in the numerator would be (1, -1*1, -3*-1*1, -5*-3*-1*1)

 

[mg0stisha]

how are you trying to explain it then?

 

[oinkbanana]

n! denotes the double factorial of n and is defined recursively for odd numbers,,
eg: 9! = 1 × 3 × 5 × 7 × 9 = 945

does that help?