(see (\[Riemann29\]), (\[Riemann30\])), the record $\sum_{i=1}^{\infty}a_i$ becomes a kind of $\sum_{i=1}^{many}a_i$.

It is worthwhile noticing also that the symbol $\infty$ cannot be used in the same expression with numerals using 1 (analogously, the record ‘many’+4 has no sense because it uses symbols from two different languages having different accuracies).

As a consequence, if one wants to work with an infinite number, $n$, of items in a sum $\sum_{i=0}^{n}
a_i$ then it is necessary to fix explicitly this number using numerals available for expressing infinite numbers in a chosen numeral system (e.g., the numeral system introduced in the previous section can be taken for this purpose). The same situation we have with sums having a finite number of items: it is not sufficient to say that the number, $n$, of items in the sum is finite, it is necessary to fix explicitly the value of $n$ using for this purpose numerals available in a chosen traditional numeral system to express finite numbers.

The new numeral system using grossone allows us to work with expressions involving infinite numbers and to obtain, where it is possible, as results infinite, finite, and infinitesimal numbers. Obviously, this is of a particular interest in connection with 