# Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$

In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area.

Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x),$$ or $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),$$ where $p(x)$ is a polynomial with all coefficients which are natural numbers, and $r_i(x)$ is a quotient of polynomials with at least one pole(which means denominator is polynomial with at least one zero point)，$r_i(x)$ can be expanded as $$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0 ?$$

Question 2: If they do exist, how to compute them?

Question 3: Under what condition $$f(x)=p(x)+\sum_{i=1}^{n}r_i(x),n <\infty?$$

Question 4: Are there finite number of $$T_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0$$,where $$r_i(x) =\sum_{j=1}^{\infty} b_{ij} x^j, \text{ }b_{ij} \in N\text{ or } b_{ij} = 0$$, are all "finitely generated" by $T_i(x)$ by multiply and addition?

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Seems to me part of the question is trivial --- if $r_i(x)=a_ix^i$ then $f(x)=\sum^{\infty}r_i(x)$. –  Gerry Myerson Jul 20 '13 at 23:56
@Gerry,maybe,but $r_i(x)$is rational quotient of polynomials,thank you for your comment, –  XL _at_China Jul 21 '13 at 0:21
@GerryMyerson You are a better man than I, since I can't even parse the question. –  Igor Rivin Jul 21 '13 at 1:15
@XL_at_China, the function $a_ix^i$ is a quotient of polynomials. It is the quotient of $a_ix^i$ and $1$, for example. –  Mariano Suárez-Alvarez Jul 21 '13 at 1:55