Find power series expansion for :

A Taylor series expansion of f(x) about point x=a can be written as:

$$f(x)=f(a)+f'(a)(x-a)+\frac{1}{2!}f''(a)(x-a)^2+...$$

If we set a = 0 here:

f(x) = (2-x)^1/2                f(0) = 2^1/2

f'(x) = -(1/2)(2-x)^-1/2      f'(0) = -(1/2)2^-1/2

f''(x) = -(1/4)(2-x)^-3/2     f''(0) = -(1/4)2^-3/2

... etc.

so

f(x) = √2 - x/(2√2) - x²/(8*2^3/2) ...  

(this could be tidied up by shifting all the √2s in the denominators to the numerators - by multiplying by √2/√2)

Similarly for the other one, which is best written as  f(x) = (1 - x²)^-1/2 to start with.

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Alan, if you are in the mood I thing that this one is for you. :)

Thank you Alan:)