3.51: Trig AP and cosec GP

The first three terms of an arithmetic progression are $-3\mathrm{tan}\dfrac{\theta}{2}$, $-\mathrm{tan}\dfrac{\theta}{2}$, $\mathrm{tan}\dfrac{\theta}{2}$, where $0<\theta<\dfrac{\pi}{2}$. Given that the $12$th term is $\dfrac{19\sqrt{3}}{3}$, find the exact value of $\theta$.

Hence find the exact sum to ten terms.

The first three terms of a geometric progression are $\dfrac{1}{16}\mathrm{cosec}^{4}z$, $\dfrac{1}{4}\mathrm{cosec}^{2}z$, $1$, where $-\dfrac{\pi}{2}<z<\dfrac{\pi}{2}$. Given that the sum of the third and fourth terms is $4$, find the possible values of $z$.

Determine whether this progression has a sum to infinity.