Are there any results to determine whether the given power series of real variable converges to a rational function? I mean just analyzing the coefficients of the series. One way is to find the sum function which is not always easy to find.

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Aiden Norman · Accepted Answer

A power series is the Taylor series of a rational function if and only if its terms satisfy a constant-coefficient linear recurrence      a    n    =      ∑          j      =      1        m        c    j        a          n      −      j      for all sufficiently large   n, where   m and       c    j   are constants.EDIT: This is certainly well known, and easier to prove than to find a printed reference. If   R  (  x  )  =  A  (  x  )      /    B  (  x  ) with   A  (  x  ) and   B  (  x  ) polynomials, then   B  (  x  )  R  (  x  )  =  A  (  x  ) gives you the recurrence       ∑    i        b    i        r          k      −      i        =  0 for   k  &amp;gt;  degree  (  A  ), where       b    i   are the coefficients of   B  (  x  ) and       r    i   the Maclaurin coefficients of   R  (  x  ). Conversely, if       ∑          j      =      0        m        c    j        a          k      −      i        =  0 for   k  &amp;gt;  n, that says   B  (  x  )  G  (  x  ) is a polynomial of degree   ≤  n where   G  (  x  ) is the generating function       ∑    k        a    k        x    k   and   B  (  x  )  =      ∑          j      =      0        m        c    j        x    j  .

Are there any results to determine whether the given power

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