Nth-term test

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n\ne 0}$ or if the limit does not exist, then ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{a}_n}$ 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:

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0,}$ then ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{a}_n}$ may or may not converge. In other words, if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0,}$ the test is inconclusive.

The harmonic series is a classic example of a divergent series whose terms approach zero in the limit as ${\displaystyle n\to \mathrm{\infty }}$. The more general class of p-series,

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p},}$

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.