Criteria for a power series representation?

  • Thread starter Master J
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  • #1
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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]

???
 
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Answers and Replies

  • #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.
 
  • #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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