Double sequence

A sequence of certain elements numbered by two indices:

$$a_{mn},\quad m,n=1,2,\ldots.$$

As compared to ordinary sequences (i.e. sequences numbered with the aid of one index), double sequences have a number of distinguishing features; thus, there are several definitions of the limit of a double sequence which are not mutually equivalent.

The concept of a numerical double sequence is closely connected with that of a numerical double series

$$\sum_{m=1}^\infty\sum_{n=1}^\infty u_{mn},$$

both the terms and the (rectangular) partial sums of such a double series,

$$S_{mn}=\sum_{k=1}^m\sum_{l=1}^nu_{kl},$$

constitute a double sequence. See also Double series.

Comments

A double sequence for which the notion of its limit causes trouble is given by, e.g.,

$$a_{mn}=\frac{m}{m+n}.$$

Letting $m\to\infty$ for $n$ fixed, and subsequently $n\to\infty$ gives as limit 1. On the other hand, letting $n\to\infty$ for $m$ fixed, and subsequently $m\to\infty$ gives 0.

See also Double limit; Repeated limit.