If we have a power series with integral coefficients $f(x)=\sum_{i=0}^{\infty}a_ix^i$ which converges for all $|x|<1$, and the limit as $x\rightarrow 1$ exists, where we approach along the real axis from the left, can we conclude $f$ is a polynomial?
The idea being that the existence of such a limit forces the sum $\sum_{i=0}^{\infty}a_i$ to converge, and integrality gives the result, but setting up this limit is proving difficult.
If it helps, we can assume $f(x)$ is positive on $[0,1)$, and dominated by a positive polynomial on the interval $[0,1]$.