Power series such that the $n$-th partial sum has $n$ distinct roots I am looking for examples of power series of the form
$$\sum_{k=0}^\infty a_k x^k$$
(where $a_k \in \mathbb{C}$ for all $k$) such that the polynomial given by its $n$-th partial sum has $n$ distinct roots, i.e.:
$$\sum_{k=0}^n a_k x^k$$
has $n$ distinct roots.
So far, I have found this family of examples: $\sum_{k=0}^\infty x^k$ and all of its derivatives. Can you help me find some examples that are not any of these?

I believe that
the exponential function
$\sum_{n=0}^{\infty} \dfrac{x^n}{n!}
$
does this.
Here is one paper
discussion the roots
of the partial sums,
although I don't see a proof
that all the roots are distinct.
https://sites.math.washington.edu/~morrow/336_09/papers/Ian.pdf
After further search,
I found this,
which does prove
that all the zeros are simple:
https://maa.tandfonline.com/doi/abs/10.1080/00029890.2005.11920265
It is the American Mathematical Monthly,
2005, vol. 10, p. 891.
