# Continuity of infinite series

## 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

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Dick
Homework Helper
|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?

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

Dick