Term test


In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series:
Many authors do not name this test or give it a shorter name.
When testing if a series converges or diverges, this test is often checked first due to its ease of use.

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:
The harmonic series is a classic example of a divergent series whose terms limit to zero. The more general class of p-series,
exemplifies the possible results of the test:
The test is typically proven in contrapositive form:
If sn are the partial sums of the series, then the assumption that the series
converges means that
for some number s. Then

Cauchy's criterion

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

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 .