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Continuity of infinite series

  • #1

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
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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|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
Hmm we haven't done differentiation yet so I'm not sure how helpful the |(f(x)-f(a))/(x-a)| will be.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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
yeah...
Thanks for the help though
 

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