Let f be a real valued function of a real variable, such that |f(n) (0)| ≤ K for all n ∈ ℕ, where K > 0. Which of the following is/are true?
1
\(\left|\frac{f^{(n)}(0)}{n!}\right|^{\frac{1}{n}} \rightarrow 0\) as n → ∞
2
\(\left|\frac{f^{(n)}(0)}{n!}\right|^{\frac{1}{n}} \rightarrow \infty\) as n → ∞
3
f(n) (x) exists for all x ∈ ℝ and for all n ∈ ℕ
4
The series \(\sum_{n=1}^{\infty} \frac{f^{(n)}(0)}{(n-1)!}\) is absolutely convergent