Summation of Zero/Indexed Summation
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Theorem
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.
Let $a, b$ be integers.
Let $\closedint a b$ denote the integer interval between $a$ and $b$.
Let $f_0 : \closedint a b \to \mathbb A$ be the zero mapping.
Then the indexed summation of $0$ from $a$ to $b$ equals zero:
- $\ds \sum_{i \mathop = a}^b \map {f_0} i = 0$
Proof
At least three proofs are possible:
- by induction, using Identity Element of Addition on Numbers
- using Indexed Summation of Multiple of Mapping
- using Indexed Summation of Sum of Mappings
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