- #1

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## Homework Statement

f(x) = [tex]\sqrt{x}[/tex], a = 4

## Homework Equations

f(x) = [tex]\sum[/tex]f[tex]^{n}[/tex](a)/n! (x-a)[tex]^{n}[/tex]

## The Attempt at a Solution

f(x) = x[tex]^{1/2}[/tex]

f[tex]^{'}[/tex](x) = [tex]\frac{1}{2}[/tex]x[tex]^{1/2}[/tex]

f[tex]^{2}[/tex](x) = -[tex]\frac{1}{2}[/tex]*[tex]\frac{1}{2}[/tex]x[tex]^{-3/2}[/tex]

f[tex]^{3}[/tex](x) = [tex]\frac{1}{2}[/tex]*[tex]\frac{1}{2}[/tex]*[tex]\frac{3}{2}[/tex]x[tex]^{-5/2}[/tex]

f[tex]^{4}[/tex](x) = -[tex]\frac{1}{2}[/tex]*[tex]\frac{1}{2}[/tex]*[tex]\frac{3}{2}[/tex]*[tex]\frac{5}{2}[/tex]x[tex]^{-7/2}[/tex]

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

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!