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

[omg4cards]

Homework Statement

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

Homework Equations

f(x) = \sumf^{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?