According to exercise 18, chp. 2, of Stein & Shakarchi's Fourier analysis, $\frac{\partial P_r(\theta)}{\partial \theta}$ is a solution to the steady-heat equation that converges only pointwise to the zero function on the boundary.
It's easy to check that it is a solution of the steady heat equation, because $\frac{\partial P_r(\theta)}{\partial \theta} = \sum_{n=0}^\infty 2inr^ne^{in\theta}$ is a linear combination of solutions ($r^ne^{in\theta}$). However, I'm having trouble to check that it converges pointwise to zero.
At $\theta=0$, I tried $$\sum_{n=0}^\infty 2inr^n=-2i\frac {r}{(1-r)^2} \quad\mbox{ for }r<1$$But I don't know if the sum on the left converges for $r=1$, and the one on the right certainly doesn't (Abel).
I thought of looking at rational $\theta$, but then the factors of $2n$ mess the cancellations.
Finally, how do I see that $\frac{\partial P_r(\theta)}{\partial \theta}$ is not zero?