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Continuity of infinite series

  1. May 12, 2008 #1
    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. jcsd
  3. May 12, 2008 #2


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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?
  4. May 13, 2008 #3
    Hmm we haven't done differentiation yet so I'm not sure how helpful the |(f(x)-f(a))/(x-a)| will be.
  5. May 13, 2008 #4


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    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.
  6. May 13, 2008 #5
    Thanks for the help though
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