Shift rule explained

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.^[1]

For series, the rule states that the series

\sum\limits ^{inf ty} _{n =1} a _{n} converges to a number if and only if \sum\limits ^{inf ty} _{n =1} a _{n+N} converges.^[2]

Notes and References

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