I've come up with an example that I think can represent every non-zero rational number:
$$\sum_{n=0}^\infty \frac{k}{mx^n}$$
where $k\in \mathbb{Z}_{\neq0}$, $m \in \mathbb{N}_{\neq0}$ and $x \in \Bbb Z \setminus \{ -1, 0, 1 \}$.
And I know there are series that converge to irrational numbers like $\sqrt{2}$ and transcendental numbers like $\pi$ or $e$, that use a finite number of variables that are non-zero integers. But is there such a series for every real number?
Edit
To be more clear, I am talking about infinite series that converge to a real number, that can be expressed in finite terms. For example $\pi$ can be expressed as
$$\sum_{n=0}^\infty \frac{4(-1)^{n}}{(2n+1)}$$
This would fit the bill as opposed to
$$\frac{3}{10^0} + \frac{1}{10^1} + \frac{4}{10^2} + \frac{1}{10^3} + \dots$$
which would require infinitely many terms to describe.
Best Answer
No, at least for most definitions of what you mean by "expressible in finite terms". There are only countably many expressions of finite length that you can write down, and there are uncountably many real numbers, so not all of them can correspond to some expression.
(Caveat: This argument only works if the kinds of "finite expressions" you allow are limited enough that you can give a precise definition of what it means for a real number to be equal to the expression. If you interpret "finite expression" broadly enough, it turns out that this is impossible because of deep logical issues related to Gödel's incompleteness theorems. See this answer on MO, for instance)