In this question some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging *to* the unit circle. Well, nothing is new under the Sun, and this is a Theorem of Jentsch, from his recent PhD thesis:

```
R. Jentsch, Untersuchungen zur Theorie der Folgen analytischer
Funktionen, Inauguraldissertation
Berlin, 1914,39 s.
```

Actually, Jentsch's theorem does not need the circle to be the natural boundary, but simply the circle of convergence, so the only thing it is lacking is the stronger statement that the empirical measure of the zeros converges to the Lebesgue measure on the unit circle.

**EXACT STATEMENT**
Let $$f(z) = \sum_{i=0}^\infty a_n z^n,$$ with radius of convergence $1.$ Then every point of the unit circle is an accumulation point of zeros of partial sums of $f.$

**END EXACT STATEMENT**

The question is: does anyone here know of a more accessible place where the proof of this result could be found (or maybe there is a 10 line argument someone could just post)?