Shift rule

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

Here ${\displaystyle n}$ and ${\displaystyle N}$ are natural numbers.

For sequences, the rule states that if ${\displaystyle (a_{n})}$ is a sequence, then it converges if and only if ${\displaystyle (a_{n+N})}$ also converges, and in this case both sequences always converge to the same number.¹

For series, the rule states that the series ${\displaystyle \sum \limits _{n=1}^{\infty }a_{n}}$ converges to a number if and only if ${\displaystyle \sum \limits _{n=1}^{\infty }a_{n+N}}$ converges.²