Shift rule

{{Short description|Mathematical rule}}

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

Here $n$ and $N$ are natural numbers.

For sequences, the rule states that if $(a\_\{n\})$ is a sequence, then it converges if and only if $(a\_\{n+N\})$ also converges, and in this case both sequences always converge to the same number.{{citation|url=http://www.ueltschi.org/teaching/2011-MA131/notes-MA131.pdf|title=Analysis –MA131|first=Daniel|last=Ueltschi|publisher=University of Warwick|year=2011|page=31}}.

For series, the rule states that the series $\backslash sum\backslash limits\_\{n=1\}^\backslash infty\; a\_\{n\}$ converges to a number if and only if $\backslash sum\backslash limits\_\{n=1\}^\backslash infty\; a\_\{n+N\}$ converges.{{citation|title=How to Think About Analysis|first=Lara|last=Alcock|authorlink= Lara Alcock |publisher=Oxford University Press|year=2014|isbn=9780191035371|page=102|url=https://books.google.com/books?id=gk5uBAAAQBAJ&pg=PA102}}.