Many authors do not name this test or give it a shorter name.[2]
When testing if a series converges or diverges, this test is often checked first due to its ease of use.
In the case of
p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-archimedean triangle inequality.
Usage
Unlike stronger
convergence tests, the term test cannot prove by itself that a series
converges. In particular, the converse to the test is not true; instead all one can say is:
If then may or may not converge. In other words, if the test is inconclusive.
The
harmonic series is a classic example of a divergent series whose terms limit to zero.[3] The more general class of
p-series,
exemplifies the possible results of the test:
If p ≤ 0, then the term test identifies the series as divergent.
The assumption that the series converges means that it passes
Cauchy's convergence test: for every there is a number N such that
holds for all n > N and p ≥ 1. Setting p = 1 recovers the definition of the statement[5]
Scope
The simplest version of the term test applies to infinite series of
real numbers. The above two proofs, by invoking the Cauchy criterion or the linearity of the limit, also work in any other
normed vector space[6] (or any (additively written) abelian group).
^For example, Rudin (p.60) states only the contrapositive form and does not name it. Brabenec (p.156) calls it just the nth term test. Stewart (p.709) calls it the Test for Divergence.