I have probably encountered hundreds of infinite series where each term is rational. In each case (as far as I can remember), the value of the infinite series was either rational or transcendental.
For example, some simple cases include:
$$\begin{align}
\sum_{r=1}^{\infty}\frac{1}{2^r}&=1\\
\sum_{r=1}^{\infty}\frac{1}{r^2}&=\frac{\pi^2}{6}\\
\sum_{r=1}^{\infty}\frac{1}{r^2+1}&=\frac{1}{2}(\pi\coth\pi-1).
\end{align}$$
I realize that it's definitely not known that it's true that infinite series of rational terms can only be rational or transcendental, as otherwise we wouldn't say that $\zeta(3)$ and other constants are irrational; we'd immediately be able to say that they are transcendental, so I'm not asking that. I'm asking if anyone knows of any infinite series of rational terms that is just irrational, not transcendental.
Thank you for your help.
Best Answer
$$\sqrt2 = 1 + \frac{4}{10} + \frac{1}{100} + \frac{4}{1000} + \frac{2}{10000} + \frac{1}{100000} + \frac{3}{1000000} + \cdots$$
In general, $$\alpha = \lfloor\alpha\rfloor + \sum_{n=1}^\infty \frac{\lfloor 10^n \alpha\rfloor - 10 \lfloor 10^{n-1}\alpha\rfloor}{10^n} $$ so every real number $\alpha$ -- be it rational, algebraic, or transcendental -- is the sum of some infinite series of rationals.