- . \[Riemann15\] Obviously, if we change the number of items, $k$, then, as it happens in the finite case, the results of summation will also change. For instance, it follows $$S_7(\G1-1)=\frac{\G1}{2}, \hspace{1cm}
 S_7(\G1+1)=\frac{\G1}{2}+1, \hspace{1cm} S_7(\G1^2)=-\frac{\G1^2}{2}.$$ In particular, for $k=\G1-1$ we have $$S_7(\G1-1) = \underbrace{1 -2   + 3 -4 + \ldots - (\G1-2) +
(\G1-1)}_{ \G1 -1\mbox{ items}}   =$$ $$\underbrace{(1     + 3   + 5 + \ldots + (\G1-3) + (\G1-1)
)}_{\frac{\mbox{\tiny{\G1}}}{2}\mbox{ items}} - \underbrace{( 2
+  4 + 6 + \ldots + (\G1-2)
)}_{\frac{\mbox{\tiny{\G1}}}{2}-1\mbox{ items}} =$$ $$\frac{(1+(\G1-1)) \G1  }{4} -
\frac{(2+(\G1-2)) (\G1/2-1)  }{2} = \frac{ \G1^2  }{4} - \left(
\frac{ \G1^2  }{4}  - \frac{\G1}{2}\right)
 =   \frac{\G1}{2}.$$ Obviously, we have (cf. (\[Riemann15\])) that $$S_7(\G1) =S_7(\G1-1)   -  \G1   = \frac{\G1}{2} -  \G1   = -
\frac{\G1}{2}.$$

Analogously to the passage from series to sums, the introduction of infinite and infinitesimal numerals allows us (in fact, it *imposes*) to substitute improper integrals of various kinds by integrals defined in a more precise way. For example, let us consider the following improper integral $$\int_{0}^{\infty}x^2 dx.$$ Now it is necessary to define its upper infinite limit of integration explicitly. Then, different infinite numbers used instead of $\infty$ will lead to different results, as it happens in 