Computing sums of conditionally convergent and divergent series using the concept of grossone

Let a1, a2, … be a numerical sequence. As the main task of the paper, we consider the classical problem of computing the sum ∑n=1∞an when the series is either conditionally convergent or divergent. We demonstrate that the concept of grossone, recently proposed by Sergeyev, can be useful in both computing this sum and studying properties of summation methods. We also consider the problem of choosing the upper limit in the sum if we wish to replace the infinity sign ∞ with a grossone-based quantity. Finally, we discuss some properties of prime numbers in the grossone universe and make an attempt of analyzing the celebrated Euler’s product formula.