sumable

Sumable is an adjective used to describe an object—typically a sequence, series, function, or array—for which the total sum can be assigned a finite value. In modern mathematical usage, the more common term is summable. The two words are often used interchangeably in nontechnical contexts, but summable is preferred in formal analysis.

In the context of series, a sequence (a_n) is called summable if the series ∑ a_n converges to a finite limit. If ∑ |a_n| converges, the sequence is absolutely summable; this stronger condition implies the convergence of ∑ a_n. When dealing with sets of nonnegative terms, summability is equivalent to the finiteness of the total sum, and many texts simply say the sum is finite.

In measure theory and functional analysis, a function f indexed by a set I is summable if the sum over I of its absolute values is finite; equivalently, the function belongs to the ℓ^1 space (or the corresponding L^1 space when integrals are involved). In discrete settings, the terms either are described as sumable or summable to describe the ability to perform a finite summation across the index set.

Common examples include: ∑_{n=1}^∞ 1/n^p is summable for p>1; the geometric series ∑ r^n is summable for |r|<1.

See also: summation, series, convergence, absolute convergence, ℓ^1 spaces.