Translation of Index Variable of Summation/Examples

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Examples of Translation of Index Variable of Summation

$\ds \sum_{1 \mathop \le j \mathop \le n} a_j = \sum_{1 \mathop \le j - 1 \mathop \le n} a_{j - 1} = \sum_{2 \mathop \le j \mathop \le n + 1} a_{j - 1}$


Proof

A direct instance of Translation of Index Variable of Summation.


Sources