Proving Continuity of Power Series Function

[ricardianequiva]

Homework Statement

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.

Homework Equations

Definition of continuity

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

 

[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?

 

[ricardianequiva]

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

 

[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.

 

[ricardianequiva]

yeah...
Thanks for the help though