Deriving the Coefficients of an Infinite Power Series

[DanAbnormal]

Homework Statement

Show that if a function f(x) can be expressed as an infinite power series, then it has the form

f(x) = f(x0) + \sum^{\infty}_{n = 1}\frac{f^{n}(x0)}{n!}(x - x0)^{}

Homework Equations

The Attempt at a Solution

I know that for an infinite power series:

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

which can be simplified into the above expression. But is there any groundwork that the question asks to get to this point here? I am thinking for 6 marsk i can't just right down the two lines...

 

[Dick]

A power series has the form <br /> f(x)= \sum^{\infty}_{n = 0}<br /> a_n (x-a)^n. You want to show f_n(a)/n!=a_n. Differentiate the power series n times and put x=a.