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?

Let $a_k$ be a sequence of numbers that are algebraically independent over the rationals.  Thus for any nontrivial polynomial $p(x_0, \ldots, x_n)$ with rational coefficients, $p(a_0, \ldots, a_n) \ne 0$.
The polynomial $P_n(x) = a_0 + a_1 x + \ldots + a_n x^n$ has a repeated root if and only if its discriminant is $0$.  In this case that discriminant is a nontrivial polynomial in $a_0, \ldots, a_n$ with integer coefficients, and therefore must be nonzero.
