# Can all real numbers be represented by the sum of a convergent series?

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.

• Well... $S_n=\sum_{i=0}^n \frac{k}{m2^i}$ works, with $\frac km=\frac q2$, with $q$ the rational you want to reach... Sep 15, 2015 at 13:10
• Every real number has a decimal expansion. A decimal expansion is a series built out of integers. What more could you want? Sep 15, 2015 at 13:23
• Your sum is trivial since if we let $x=2$, we get $\sum_{n=0}^\infty\frac{1}{2^n}=2$. Hence you can choose $k$ and $m$ such that $\frac{k}{m}\cdot 2=\ell$, where $\ell\in\mathbb{Q}$ is any rational you like. Sep 15, 2015 at 13:53
• You should use digits for a natural summation that converges to a real. Sep 15, 2015 at 14:21
• @Marijn Could you example series for $\sqrt{2}$, $\pi$, and $e$? It seems like everyone, including yours truly, is confused about exactly what kinds of series you are allowing. Sep 15, 2015 at 16:20

• You could specify exactly what operations you're allowed to use when writing down a formula for the $n$th term in the series (e.g., arithmetic operations, factorials, integer constants). Writing this out as a rigorous definition would take a bit of work and would involve some kind of inductive definition (see the discussion here, for instance). Sep 16, 2015 at 7:56