In mathematics, the nth-term test for divergence[1] is a simple test for the divergence of an infinite series:
IfMany authors do not name this test or give it a shorter name.[2]or if the limit does not exist, then\limnan ≠ 0
diverges.
infty \sum n=1 an
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.
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:
IfThe harmonic series is a classic example of a divergent series whose terms approach zero in the limit asthen\limnan=0,
may or may not converge. In other words, if
infty \sum n=1 an
the test is inconclusive.\limnan=0,
n → infty
| infty | |
| \sum | |
| n=1 |
| 1 | |
| np |
,
exemplifies the possible results of the test:
The test is typically proven in contrapositive form:
Ifconverges, then
infty \sum n=1 an
\limnan=0.
If sn are the partial sums of the series, then the assumption that the seriesconverges means that
\limn\toinftysn=L
\limn\toinftyan=\limn\toinfty(sn-sn-1)=\limn\toinftysn-\limn\toinftysn-1=L-L=0.
Assuming that the series converges implies that it passes Cauchy's convergence test: for every
\varepsilon>0
\left|an+1+an+2+ … +an+p\right|<\varepsilon
\limn\toinftyan=0.
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.