What does summability mean?
Summability is the mathematical property of a sequence or series having a sum. For an infinite series, this normally means that the successive partial sums converge to a finite limit. The term matters because the presence of infinitely many terms does not by itself prevent a series from having a meaningful total. A series may be summable while its individual terms approach zero, but that condition alone is not sufficient to establish ordinary summability. In more advanced work, “summability” can also refer to the possibility of assigning a value to a divergent series through a specified summation method. Outside mathematics, the word is uncommon, and its form should not be confused with summation, which names the process of adding terms or the resulting total.
nounThe property of a sequence or series of terms being capable of being assigned a sum, especially when the terms are infinite and the convergence of the series must be established.
- The property of an infinite sequence or series being capable of having a sum assigned to it, particularly through convergence of its partial sums.
"The summability of the series depends on whether its partial sums converge to a finite limit."
"The theorem provides a sufficient condition for the summability of the series."
"Different summation procedures can give different answers when ordinary summability does not hold."
Rarely pluralized; the word is normally treated as an abstract mass noun.
"The summabilities of these series depend on the convergence criteria being used."
Not every infinite series has a sum, however quickly its terms shrink.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does summability come from?
Summability was formed in English from the stem sum together with the suffixal elements associated with able and ability, expressing the abstract property of being sumable. Its ultimate root is the English sum, which comes through Latin summa, a feminine form of sum, meaning a total or amount. The same Latin foundation also underlies English words such as summation and, in a separate semantic development, consummate.
Related word forms
How summability is actually used
This is primarily technical mathematical vocabulary. It is used especially for infinite series and may refer either to ordinary convergence or to summability under a specified generalization of ordinary summation.
Easily confused with summability
Summability is the property of being capable of being summed, whereas summation is the act or result of adding terms together.