<P> Note that we can consider multiple sequences at the same time by using different variables; e.g. (b n) n ∈ N (\ displaystyle (b_ (n)) _ (n \ in \ mathbb (N))) could be a different sequence than (a n) n ∈ N (\ displaystyle (a_ (n)) _ (n \ in \ mathbb (N))). We can even consider a sequence of sequences: ((a m, n) n ∈ N) m ∈ N (\ displaystyle ((a_ (m, n)) _ (n \ in \ mathbb (N))) _ (m \ in \ mathbb (N))) denotes a sequence whose mth term is the sequence (a m, n) n ∈ N (\ displaystyle (a_ (m, n)) _ (n \ in \ mathbb (N))). </P> <P> An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values . For example, the notation (k 2) k = 1 10 (\ displaystyle (k ^ (2)) _ (k = 1) ^ (10)) denotes the ten - term sequence of squares (1, 4, 9,..., 100) (\ displaystyle (1, 4, 9,..., 100)). The limits ∞ (\ displaystyle \ infty) and − ∞ (\ displaystyle - \ infty) are allowed, but they do not represent valid values for the index, only the supremum or infimum of such values, respectively . For example, the sequence (a n) n = 1 ∞ (\ displaystyle (a_ (n)) _ (n = 1) ^ (\ infty)) is the same as the sequence (a n) n ∈ N (\ displaystyle (a_ (n)) _ (n \ in \ mathbb (N))), and does not contain an additional term "at infinity". The sequence (a n) n = − ∞ ∞ (\ displaystyle (a_ (n)) _ (n = - \ infty) ^ (\ infty)) is a bi-infinite sequence, and can also be written as (..., a − 1, a 0, a 1, a 2, ...) (\ displaystyle (..., a_ (- 1), a_ (0), a_ (1), a_ (2), ...)). </P> <P> In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off . That is, one simply writes (a k) (\ displaystyle (a_ (k))) for an arbitrary sequence . Often, the index k is understood to run from 1 to ∞ . However, sequences are frequently indexed starting from zero, as in </P> <Dl> <Dd> (a k) k = 0 ∞ = (a 0, a 1, a 2, ...). (\ displaystyle (a_ (k)) _ (k = 0) ^ (\ infty) = (a_ (0), a_ (1), a_ (2), ...).) </Dd> </Dl>

How to indicate the start of a formula in numbers