I am currently working on the "Abel's angular convergence theorem", which I recall the statement here :
Theorem : Let $f(z)=\sum_{n=0}^{+\infty} a_n z^n$ be a power series of radius of convergence equal to $1$. Let $\theta_0 \in \left[ 0,\frac{\pi}{2}\right[$ and define $$\Delta_{\theta_0} = \lbrace z \in D(0,1) \text{ }| \text{ } \exists \rho > 0, \theta \in \left[- \theta_0, \theta_0 \right], z=1-\rho e^{i\theta}\rbrace$$ If the series $\sum_{n \geq 0} a_n$ converges, then $$\lim_{z \rightarrow 1 \\ z \in \Delta_{\theta_0}} f(z) = \sum_{n =0}^{+\infty} a_n.$$
My question is simple : do you know an application of this theorem ? In particular, do you know an application where the "angular" part is used ?
Everytime this theorem is mentionned, I read as an application that you can deduce the relations $$\sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{n} = \ln(2), \quad \quad \quad \quad \text{or} \quad \quad \quad \quad \sum_{n=0}^{+\infty} \frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}$$
but I think these examples are really bad because you don't need this theorem for them, since the series $\sum \frac{(-1)^{n+1}}{n} x^n$ and $\sum_{n=0}^{+\infty} \frac{(-1)^{n}}{2n+1}x^n$ are clearly uniformly convergent over $[0,1]$.
So, does someone know a case where this theorem is really needed ?
Thanks a lot !
