Convergence of series $T_z$

If we take the Ratio Test of $T_{z}$ :

$\frac{T_{z+1}}{T_{z}} = \frac{2(z+1)-1}{2(z+1)+1} \cdot \frac{2z+1}{2z-1} = \frac{4z^2 + 4z +1}{4z^2 + 4z -3}$ ;

and $| \frac{4z^2 + 4z +1}{4z^2 + 4z -3} | > 1$ as $z \rightarrow \infty$ . Hence the sequence is divergent.