Doubly infinite series is a series $a_n$ where $n$ ranges from $-\infty$ to $\infty$.

Warning

Doubly infinite series is a term which should be used with caution, as some people use the term for the double series (series with two indices), so that it is an infinite series in each variable, say $\sum_{i=1}^\infty\sum_{j=1}^\infty a_{i j}$.

Definition

Doubly infinite series

$\sum_{k = -\infty}^\infty a_k,$

is a pair of two (ordinary infinite) series, its negative part $\sum_{k = -1}^\infty a_{-k}$ and its positive part $\sum_{k = 0}^\infty a_k$ (one could say nonnegative part, but it is not customary). A doubly infinite series converges if each of its constituent parts (positive and negative), converges; the sum of the series is then, by definition, the sum of the limits of the two series.

Notion to cover later: principal value of a double series