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I was learning more about series today and would like to know if there are existing proofs I could look at about this problem. Basically, if you are given an infinite series representing a function f : $\Bbb N \Rightarrow \Bbb R$ but only shown the first n numbers, how many functions f, written in terms of n, could you write to represent that series. I'm not including piecewise functions, because I assume that would always be infinite.

Take the series $(2, 4, ...)$ with 2 numbers given. $f(n)=2n$ , $f(n)=n^2-n+2$ , and $f(n)=2^n$ would all be functions that could fit this series, although they differ after the first two numbers. I believe there are more polynomials that fit this description but I'm not sure how many.

My question is, essentially, are there an infinite number of functions for which $f(1) = 2$ and $f(2) = 4$, and if this is the case, does this also apply to any finite number of outputs? (e.g. the first n digits of pi written as $(3, 1, 4, 1, 5, 9...)$) If not, could you find out how many possible functions there are?