Is it possible to express all transcendental numbers (and more generally all real numbers $\in \mathbb R$) as the sum of an infnite series of the form
$$\sum_{n=0}^{∞} \frac{p(n)}{q(n)}$$ where $p(n)$ and $q(n)$ are polynomials with rational coefficients (we assume that the degree of $p$ and $q$ are integers and that they differ by at least $2$, which is required for convergence ).
My approach to this problem would be to show that the set of algebraic numbers forms a countable set, and that the set of rational functions $\frac{p(n)}{q(n)}$ would also be countable, so the transcendental numbers would form a countable set — but they are known to form an uncountable set, a contradiction. This would indicate that not all transcendental can be expressed by the following series.
Any ideas for a formal proof? Was my approach correct? Thanks for help.
Best Answer
Yes, your suggestion works: more generally, given any countable set $\mathcal{F}$ of functions, only countably real numbers can have the form $\sum_{n\in\mathbb{N}}f(n)$ for some $f\in\mathcal{F}$ so "most" real numbers won't be expressible in such a way (here $\mathcal{F}$ is the set of rational functions with rational coefficients).