Indexed Summation over Interval of Length One
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Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.
Let $a \in \Z$ be an integer.
Let $f: \set a \to \mathbb A$ be a mapping on the singleton $\set a$.
Then the indexed summation:
- $\ds \sum_{i \mathop = a}^a \map f i = \map f a$
Proof
We have:
\(\ds \sum_{i \mathop = a}^a \map f i\) | \(=\) | \(\ds \sum_{i \mathop = a}^{a - 1} \map f i + \map f a\) | Definition of Indexed Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + \map f a\) | Definition of Indexed Summation, $a - 1 < a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f a\) | Identity Element of Addition on Numbers |
$\blacksquare$