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Math Help - power series convergence

  1. #1
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    power series convergence

    i was reading a proof on :
    given that the power series converges on (-R,R) then it is continuous on the open interval(-R,R).

    i know that i have to split it up into using triangular inequality where for all ε>0
    let
    δ'>o st l f(x) - f_m(x) l < ε/3

    and

    δ'' >0 st l f_m(x) - f_m(c) l < ε/3 ( continous at c).

    im wondering, does δ' and δ'' depend on ε or x?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    This is a relatively difficult theorem. Its proof pretty much goes along the lines of the proof of Abel's Theorem.

    It's not true for series of functions in general; for example, trigonometric series may be discontinuous. This is something special about power series.
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  3. #3
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    does δ depend on ε in the proof?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    It very well can, yes, as for any proof that a function is continuous at a point.
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