I was reading this paper by L.J. Mordell (original paper by Morris Newman). I am trying to apply first theorem from the paper with $f(x)=x$ and $k=\pi$. So the series $\displaystyle \sum_{n=0}^{\infty} \lfloor n\pi \rfloor x^n$ must not converge to a rational function for $|x|<1$.
But it seems that the series converges to a rational function as the series can be written as $3x+6x^2+9x^3+...=\frac{3x}{(x-1)^2}$ after we evaluate $\lfloor n\pi \rfloor$ for each $n$? Am I doing something wrong with the evaluation of $\lfloor n\pi \rfloor$?
EDIT: David found my mistake. Now the question is can we find the sum of this series or show that the sum is not a rational function without Mordell's result?