# Definition:Termial

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## Contents

## Definition

Let $n \in \Z_{> 0}$ be a positive integer.

The **termial** of $n$ is denoted $n?$ and defined as:

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

## Extension to Real Numbers

Let $x \in \R$ be a real number.

The **termial** of $x$ is denoted $x?$ and defined as:

- $\displaystyle x? = \dfrac {x \left({x + 1}\right)} 2$

## Example

### Termial of $\frac 1 2$

The termial of $\dfrac 1 2$ is given by:

- $\dfrac 1 2 ? = \dfrac 3 8$

## Historical Note

The **termial** was invented by Donald E. Knuth in his *The Art of Computer Programming*.

It was devised as an analogy of the factorial, so as to illustrate the extension of the latter to the real numbers.

## 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: $(9)$