Nth-term Test

In mathematics, the ''n''th-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:

If $\lim_ a_n \neq 0$ or if the limit does not exist, then $\sum_^\infty a_n$ diverges.

Many authors do not name this test or give it a shorter name.For example, Rudin (p.60) states only the contrapositive form and does not name it. Brabenec (p.156) calls it just the ''n''th term test. Stewart (p.709) calls it the Test for Divergence. Spivak (p.473) calls it the Vanishing Condition.

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:

If $\lim_ a_n = 0,$ then $\sum_^\infty a_n$ may or may not converge. In other words, if $\lim_ a_n = 0,$ the test is inconclusive.

The harmonic series is a classic example of a divergent series whose terms approach zero in the limit as $n \rightarrow \infty$. The more general class of ''p''-series,
:$\sum_^\infty \frac,$

exemplifies the possible results of the test:
* If ''p'' ≤ 0, then the ''n''th-term test identifies the series as divergent.
* If 0 < ''p'' ≤ 1, then the ''n''th-term test is inconclusive, but the series is divergent by the integral test for convergence.
* If 1 < ''p'', then the ''n''th-term test is inconclusive, but the series is convergent by the integral test for convergence.

Proofs

The test is typically proven in contrapositive form:

If $\sum_^\infty a_n$ converges, then $\lim_ a_n = 0.$

Limit manipulation

If ''s''_''n'' are the partial sums of the series, then the assumption that the series
converges means that
:$\lim_ s_n = L$
for some number ''L''. Then
:$\lim_ a_n = \lim_(s_n-s_) = \lim_ s_n - \lim_ s_ = L-L = 0.$

Cauchy's criterion

Assuming that the series converges implies that it passes Cauchy's convergence test: for every $\varepsilon>0$ there is a number ''N'' such that

:$\left, a_+a_+\cdots+a_\<\varepsilon$
holds for all ''n'' > ''N'' and ''p'' ≥ 1. Setting ''p'' = 1 recovers the claim
:$\lim_ a_n = 0.$

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 spaceHansen p.55; Șuhubi p.375 or any additively written abelian group.

Notes

References

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{{Calculus topics

Convergence tests
Articles containing proofs