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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?

MathematicalTrex
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Yes, there are always infinitely many such functions. Even if we restrict attention to "reasonably nice" functions, the answer is still yes (unless we make a truly extreme restriction). For example, we can always find infinitely many polynomials; this is a consequence of the fact that for any sequence $a_1,...,a_n$ there is a polynomial satisfying $f(1)=a_1,...,f(n)=a_n$ (both this result and its application to your question are good exercises).

For an example of a restriction which does allow us to determine $f$ from finitely much information, consider something like "polynomials of degree $\le k$" for some fixed $k$. Here it is indeed the case that from enough values (specifically, $k+1$) we can determine a unique appropriate $f$. But this is an extremely strong restriction.

Noah Schweber
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