Shift 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.

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