Term test

In mathematics, the nth-term test for divergence[1] is a simple test for the divergence of an infinite series:

Many authors do not name this test or give it a shorter name.[2]


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.[3] The more general class of p-series,

exemplifies the possible results of the test:


The test is typically proved 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 s. Then[4]

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[5]


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).


  1. Kaczor p.336
  2. 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.
  3. Rudin p.60
  4. Brabenec p.156; Stewart p.709
  5. Rudin (pp.59-60) uses this proof idea, starting with a different statement of Cauchy criterion.
  6. Hansen p.55; Șuhubi p.375


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