The shift rule is a mathematical rule for

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

s and

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

. Here

n

and

N

are

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

. For sequences, the rule states that if

(a_)

is a sequence, then it

converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) app ...

s if and only if

(a_)

also converges, and in this case both sequences always converge to the same number. For series, the rule states that the series

\sum\limits_^\infty a_

converges to a number if and only if

\sum\limits_^\infty a_

converges..

References

{{DEFAULTSORT:Shift Rule Sequences and series