Shift rule

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

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 ${\Sigma^{\infty}_{n=1}}(a_{n})$ converges to a number if and only if ${\Sigma^{\infty}_{n=1}}(a_{n+N})$ converges.^[2]

References

1. Ueltschi, Daniel (2011), Analysis –MA131 (PDF), University of Warwick, p. 31.
2. Alcock, Lara (2014), How to Think About Analysis, Oxford University Press, p. 102, ISBN 9780191035371.