the Riemann zeta function. For instance, it becomes possible to reconsider the result (\[1+1\]) that is very difficult to be fully realized by anyone who is not a mathematician. In fact, when one has an infinite sum of positive finite integers he or she would expect to have an infinite positive integer as a result. In contrast, (\[1+1\]) proposes a negative finite fractional number as the answer.

In order to become acquainted with the way the new methodology is applied to the theory of divergent series, let us consider several example. We start by considering two infinite series $S_1=7+7+7+7+\ldots$ and $S_2=3+3+3+3+\ldots$ The traditional analysis gives us a very poor answer that both of them diverge to infinity. Such operations as, e.g., $\frac{S_2}{S_1}$ and $S_2 -
S_1$ are not defined.

The new mathematical language using grossone allows us to indicate explicitly the number of their items. Suppose that the series $S_1$ has $k$ items and $S_2$ has $n$ items: $$S_1(k)=\underbrace{7+7+7+\ldots+7}_{k \mbox{ items}},
\hspace{1cm} S_2(n)=\underbrace{3+3+3+\ldots+3}_{n \mbox{
items}}.$$ Then $S_1(k)=7k$ and $S_2(n)=3n$ and by giving different numerical values (finite or infinite) to $k$ and $n$ we obtain different numerical values for the sums. For chosen $k$ and $n$ it becomes possible to calculate 