In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series: diverges.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.

In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-Archimedean ultrametric 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: the test is inconclusive.The harmonic series. The more general class of p-series,

exemplifies the possible results of the test:

• If p ≤ 0, then the nth-term test identifies the series as divergent.

• If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.

• If 1 < p, then the nth-term test is inconclusive, but the series is convergent by the integral test for convergence.

Proofs

The test is typically proven in contrapositive form:

Limit manipulation

If sn are the partial sums of the series, then the assumption that the series converges means that

for some number L. Then

Cauchy's criterion

Assuming that the series converges implies that it passes Cauchy's convergence test there is a number N such that

holds for all n > N and p ≥ 1. Setting p = 1 recovers the claim

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 or any additively written abelian group.

Notes

References