How much information we can get about the function $f: [0, R[ \to \mathbb{R} : x \mapsto \sum_{ n = 0}^\infty a_nx^n$ just by looking at the sequence $(a_n)_{n \in \mathbb{N}}$ ?
For example is it possible to get a good approximation on the shape of the graph of $f$ ? Is it possible to get information about convexity/concavity of the function ?
Also is it possible to get a good approximation of where the local extrema/minima of $f$ are situated in $[0, R[$ ?
I really want to know if all of this can be find just by looking at the sequence $(a_n)_{n \in \mathbb{N}}$ ? So you are not allowed to explicitly compute the function $f$.
For example it's well known that
$$\ln(1-x) = -\sum_{n = 1}^\infty \frac{1}{n} x^n.$$
If I just give you the sequence $a_n = -\frac{1}{n}$, is it possible to get a lot of information about $x \mapsto -\sum_{n = 1}^\infty \frac{1}{n}x^n$ so that it's possible to say it's quite logical that this sequence encoded the function $\ln(1-x)$ ?
Thank you !