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 $a_i = ua_j + v$.
EDIT: Pardon me, but it has been shown in the comments by robjohn and Michael that these are not linearly independent. Indeed:$$91a_1-10a_2=10$$&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;— Akiva Weinberger


Think of a series of real numbers with decimal expansions like

0.1100110000110000001100000000110000000000110000000...
0.0011001001001000010010000001001000000001001000000...
0.0000000110000100100001000010000100000010000100000...
0.0000000000000011000000100100000010000100000010000...
0.0000000000000000000000011000000001001000000001000...
0.0000000000000000000000000000000000110000000000100...
0.0000000000000000000000000000000000000000000000011...
...


That is, a given digit is only 1 in one the numbers in the series, and 0 everywhere else, and distributed like the above.

All those numbers are irrational because their decimal expansion never repeats, they are linearly independent, and their sum is 1/9 = 0.111111...


  EDIT: Ángel Valencia proposes the following, unfortunately also without proof. It seems likely to work to me, but I (RemcoGerlich) am working on my own fix with proof.


0.10010000000100000000000000100000000000000000000001000000...
0.01101100011011000000000011011000000000000000000110110000...
0.00000011100000111000011100001110000000000000111000001110...
0.00000000000000000111100000000001111000001111000000000000...
0.00000000000000000000000000000000000111110000000000000000...
...


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;—
There is a theorem (by Riemann, I think) which states that:
"A convergent series which is not absolutely convergent
can be rearranged to sum to any number we choose"    

So all we need to do to show that what the OP asks is possible, is to take
an arbitrary convergent series which is not absolutely convergent,
and which consists of irrational numbers only.  

Then by applying the theorem, it means we can rearrange its elements
and get any rational (and also any irrational) number we want.       

An example of such a series is, let's say:       

$\sum_\limits{n=1}^\infty \frac{(-1)^n}{n\sqrt5}$