Sum of Summations equals Summation of Sum/Infinite Sequence
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Theorem
Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers $\Z$.
Let $\displaystyle \sum_{R \paren i} x_i$ denote a summation over $R$.
Let the fiber of truth of $R$ be infinite.
Let $\displaystyle \sum_{R \paren i} b_i$ and $\displaystyle \sum_{R \paren i} c_i$ be convergent.
Then:
- $\displaystyle \sum_{R \paren i} \paren {b_i + c_i} = \sum_{R \paren i} b_i + \sum_{R \paren i} c_i$
Proof 1
Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.
Then:
\(\displaystyle \sum_{R \left({i}\right)} \left({b_i + c_i}\right)\) | \(=\) | \(\displaystyle \sum_{R \left({i}\right)} \left({a_{i 1} + a_{i 2} }\right)\) | by definition | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{R \left({i}\right)} \left({\sum_{1 \mathop \le j \mathop \le 2} a_{i j} }\right)\) | Definition of Summation | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{1 \mathop \le j \mathop \le 2} \left({\sum_{R \left({i}\right)} a_{i j} }\right)\) | Exchange of Order of Summation: Finite and Infinite Series | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{R \left({i}\right)} a_{i 1} + \sum_{R \left({i}\right)} a_{i 2}\) | Definition of Summation | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{R \left({i}\right)} b_i + \sum_{R \left({i}\right)} c_i\) | by definition |
$\blacksquare$
Proof 2
By definition, $\displaystyle \sum_{R \paren i} b_i$ and $\displaystyle \sum_{R \paren i} c_i$ are sequences in $\R$.
Hence the result as an instance of Sum Rule for Real Sequences.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$