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  Operational definition A “formal power series” is an infinite sequence of numbers (a_0, a_1, a_2,...), which we operate on by certain formal rules that are easier to remember (and whose motivation is clearer) if we represent the sequence by the symbol a_0 + a_1 x + a_2 x^2 +... 
  For instance, the “formal derivative” of the sequence (a_0, a_1, a_2, a_3,...) is defined as (a_1, 2 a_2, 3 a_3,...). 
  With this notion of convergence, the ring of formal power series becomes a “topological ring”, once we have proved the following theorem: Theorem: The operations of addition and multiplication respect the notion of convergence. 
 www.math.harvard.edu /~propp/192/0918.doc (1696 words) 
