The shift rule is a mathematical rule for sequences and series. Here n {\displaystyle n} and N {\displaystyle N} are natural numbers. For sequences, the rule states that if ( a n ) {\displaystyle (a_{n})} is a sequence, then it converges if and only if ( a n + N ) {\displaystyle (a_{n+N})} also converges, and in this case both sequences always converge to the same number. For series, the rule states that the series ∑ n = 1 ∞ a n {\displaystyle \sum \limits _{n=1}^{\infty }a_{n}} converges to a number if and only if ∑ n = 1 ∞ a n + N {\displaystyle \sum \limits _{n=1}^{\infty }a_{n+N}} converges.
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