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Criteria for a power series representation?

  1. Sep 12, 2012 #1
    I've used many different power series representations of functions and seem to always take it for granted that functions which are "nice" and continuous have such a representation.

    But what is the criteria for a function to have a power series representation? I know of some that don't, but how can one tell if a function can be represented as such?

    EDIT:

    I may as well ask, is there a proof or derivation for the power series identity

    f(x) = [itex]\sum[/itex][itex]^{\infty}_{n=0}[/itex] a[itex]_{n}[/itex] x[itex]^{n}[/itex]

    ???
     
    Last edited: Sep 12, 2012
  2. jcsd
  3. Sep 12, 2012 #2

    mathman

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    Assuming the power series is correct for small enough x, then a radius of convergence (r) can be defined where f(x) = power series for |x| < r.
     
  4. Sep 12, 2012 #3

    Stephen Tashi

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    As mathman suggests, the question of whether f(x) can be expanded as a power series is not a yes-or-no question. Some power series converge for all values of x; some power series only converge for particualr values of x.

    In the wikipedia article on analytic functions (http://en.wikipedia.org/wiki/Analytic_function) under the section "alternate characterizations", conditions are given for a real valued function to be expandable in a power series for all points x in an open set D.
     
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