Using basic tools of mathematical analysis and elementary probabilitytheory we address several problems on the irrationality of series of distinctunit fractions, \(\sum_k 1/a_k\) . In particular, we study subseries of the Lambert series \(\sum_k 1/(t^k-1)\) and two types of irrationality sequences \((a_k)\) introduced by PaulErdős and Ronald Graham. Next, we address a question of Erdős, who askedhow rapidly a sequence of positive integers \((a_k)\) can grow if both series \(\sum_k 1/a_k\) and \(\sum_k 1/(a_k+1)\) have rational sums. Our construction of double exponentiallygrowing sequences \((a_k)\) with this property generalizes to any number \(d\) of series \(\sum_k 1/(a_k+j)\) , \(j=0,1,2,\ldots,d-1\) ,and, in particular, also gives a positive answerto a question of Erdős and Ernst Straus on the interior of the set of \(d\) -tuples of their sums.Finally, we prove the existence of a sequence \((a_k)\) such that all well-defined sums \(\sum_k 1/(a_k+t)\) , \(t\in\mathbb{Z}\) , are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.
