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 $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges to a number if and only if $\sum _{n=1}^{\mathrm{\infty }}{a}_{n+N}$ converges.