Termial on Real Numbers is Extension of Integers

Theorem

The termial function as defined on the real numbers is an extension of its definition on the integers $\Z$.

Proof

From the definition of the termial function on the integers:

$\ds n? = \sum_{k \mathop = 1}^n k = 1 + 2 + \cdots + n$

From Closed Form for Triangular Numbers, we have that:

$\ds \forall n \in \Z_{> 0}: \sum_{k \mathop = 1}^n k = \dfrac {n \paren {n + 1} } 2$

This agrees with the definition of the termial function on the real numbers.

Hence the result, by definition of extension.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(10)$