# Continuity of infinite series

1. May 12, 2008

### ricardianequiva

1. The problem statement, all variables and given/known data
Show, from the definition of continuity, that the power series function f(x)=sum(a_n*x^n) is continuous for its radius of convergence.

2. Relevant equations
Definition of continuity

3. The attempt at a solution
Must show that for any |a| < R, given e>0 there exists d>0 such that |x-a|<d => |f(x) - f(a)|.
|f(x)-f(a)| < e.
|f(x) - f(a)| <= |f(x-a)|
Then I get stuck here.
Any help would be appreciated

Last edited: May 12, 2008
2. May 12, 2008

### Dick

|f(x)-f(a)| is not less than |f(x-a)|. It's not like f is linear or something. |f(x)-f(a)|=|(f(x)-f(a))/(x-a)|*|x-a|. Now to get a d, you need a bound on |(f(x)-f(a))/(x-a)| near x=a. Hint: doesn't that look like a difference quotient?

3. May 13, 2008

### ricardianequiva

Hmm we haven't done differentiation yet so I'm not sure how helpful the |(f(x)-f(a))/(x-a)| will be.

4. May 13, 2008

### Dick

You are doing power series without having done differentiation!??? That's an interesting pedagogical approach. You can still factor (x-a) algebraically from each power of f(x)-f(a), but I'm not sure how you show the rest of it converges without using the differentiability of power series.

5. May 13, 2008

### ricardianequiva

yeah...
Thanks for the help though