This is the problem 6.19 from the book Distributions and Operators, Gerd Grubb. I already have done parts (a) and (b).
The part (a) of this problem is proving that for $r\in(0,1]$, the sequence $$\{\frac{1}{2\pi}\sum_{n=-N}^{N}r^{|n|}e^{inx}\}_{N\in\mathbb{N}}$$ converges to a distribution $P_{r}$ in $D'((-\pi,\pi))$ and that $P_{1}=\delta$. Part (b) is just showing that $r\mapsto P_{r}$ is continuous.
Now, for part (c), I have to show that when $r$ converges to $1$ from the left, then $$\int_{-\pi}^{\pi}\frac{1-r^2}{1-2r\cos\theta+r^2}\;\varphi(\theta)\;d\theta$$ converges to $\varphi(0)$ for any $\varphi\in C_{0}^{\infty}((-\pi,\pi))$.
I thought about this for a while but I don't have a clue. Thanks for the help.