Does for any sequence $(a_n)$ exist a power series $f$ such that $f(i)=a_i$? Consider the sequence of real numbers $(a_n)$. Does there exist power series $f(x)=\sum_{i=0}^\infty c_ix^i$, where $c_i \in \mathbb R$, such that $f(k)=a_k$ or $f(kx_0)=a_k$ for some $x_0>0$?
I know that the answer is true for holomorfic function $f$ (according to Analytic "Lagrange" interpolation for a countably infinite set of points?), but is it true for $f$ is power series or even polynomial?
 A: Consider a Newton-style interpolation formula:
$$f(x) = b_0 + b_1 x (x/1)^{n_1} + b_2 x (x-1) (x/2)^{n_2} + b_3 x (x-1) (x-2) (x/3)^{n_3} + \cdots.$$
By recursion, we can find $b_0$ which guarantees $f(0) = a_0$, then find $b_1$ which guarantees $f(1) = a_1$ and choose $n_1$ according to some criteria below, then find $b_2$ which guarantees $f(2) = a_2$ and choose $n_2$ by the below criteria, etc.  Furthermore, within this process, we can choose $n_1, n_2, n_3, \ldots$ such that none of the terms in the expansion "overlap" in terms of producing multiple coefficients for any $x^i$; and we can also choose $n_1, n_2, n_3, \ldots$ such that the coefficients of the resulting power series will satisfy the requirements of the Root Test and give an infinite radius of convergence.  (In particular, if one of the terms of the expansion of $b_i x(x-1)(x-2)\cdots(x-i+1)$ is $C x^m$ then multiplying by $(x/i)^{n_i}$ gives $C/i^{n_i} x^{m+n_i}$, so the root test term would give $|C|^{1/(m+n_i)} / i^{n_i/(m+n_i)}$ which approaches $1/i$ as $n_i \to \infty$, so by choosing $n_i$ sufficiently large, we can ensure all of these terms are less than $2/i$.)
A: It is not possible to find  a polynomial (as mentioned in the comments) but you can find an entire function withe stated property. A much stronger result is proved in Rudin's RCA. See "An Interpolation Problem" in the chapter on "zeros of Holomorphic Functions". [ Take $A=\{0,1,2...\}$ and $m(k)=0$ for all $k$ in that theorem].
