Shift rule

Short description: Shift rule

The shift rule is a mathematical rule for sequences and series.

Here [math]\displaystyle{ n }[/math] and [math]\displaystyle{ N }[/math] are natural numbers.

For sequences, the rule states that if [math]\displaystyle{ (a_{n}) }[/math] is a sequence, then it converges if and only if [math]\displaystyle{ (a_{n+N}) }[/math] also converges, and in this case both sequences always converge to the same number.

For series, the rule states that the series [math]\displaystyle{ \sum\limits_{n=1}^\infty a_{n} }[/math] converges to a number if and only if [math]\displaystyle{ \sum\limits_{n=1}^\infty a_{n+N} }[/math] converges.