Sequence of real values as derivatives of a function.

Question

Given a sequence of real numbers $a_1, a_2, ...$, I am looking to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is infinitely differentiable at $0$ and $f^{(n)}(0)=a_n$. Edit: I'd imagine that this is easier with certain restrictions on the sequence, like if we assume that $\{a_n\}$ cycles through finitely many values.

I would like to use this function to study infinite sequences. Also, I am not expecting a complete solution to my question, I'm just looking for a starting direction to look. Thanks in advance!

Answer 1

Consider the power series $$f(x)=\sum_{i=1}^\infty \frac{a_i}{i!}x^i$$ if it converges.

Hope it helps:)

Answer 2

Borel's Theorem in its simplest form says that, given any sequence $(a_n)_{n\in\mathbb{N}_0}$, there is an infinitely often differentiable function $f\colon\mathbb{R}\to\mathbb{R}$ such that $f^{(n)}(0)=a_n$ for all $n\in\mathbb{N}_0$.