For every n ≥ 1, consider the entire function \(p_n(z)=\sum_{k=0}^n \frac{z^k}{k !}\). Which of the following statements are true?
1
The sequence of functions (pn)n ≥ 1 converges to an entire function uniformly on compact subsets of ℂ.
2
For all n ≥ 1, pn has a zero in the set {z ∈ ℂ : |z| ≤ 2023}.
3
There exists a sequence (zn) of complex numbers such that \(\displaystyle \lim _{n \rightarrow \infty}\left|z_n\right|=\infty\) and pn(zn) = 0 for all n ≥ 1.
4
Let Sn denote the set of all the zeros of pn. If \(\displaystyle a_n=\min _{z \in S_n}|z|\), then an → ∞ as n → ∞.