Can an infinite sum of irrational numbers be rational? Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear combination of the other ?
By linear combination, we mean there exists some rational numbers $u,v$ such that $a_i = ua_j + v$.

Yes, for example $\tan(\frac{\pi}{4})=1$
Write $\tan$ as an infinite Taylor expansion in powers of $\pi$.
Other trigonometric functions of irrational arguments can also have rational outputs, eg.           $\sin(\frac{\pi}{6}) = \frac{1}{2}$.
