On the dimension of graded algebras

To each graded algebra $R$ with a finite number of generators we associate the series $T(R,z)=\sum d_nz^n$, where $d_n$ is the dimension of the homogeneous component of $R$. It is proved that if the dimensions $d_n$ have polynomial growth, then the Krull dimension of $R$ cannot exceed the order of the pole of the series $T(R,z)$ for $z=1$ by more than 1.