# Infinite power series

1. Aug 19, 2008

### DanAbnormal

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

3. 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? Im thinking for 6 marsk i cant just right down the two lines...

2. Aug 19, 2008

### Dick

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