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Term test

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

Contents

  • If lim n → ∞ a n ≠ 0 or if the limit does not exist, then ∑ n = 1 ∞ a n diverges.

    • Many authors do not name this test or give it a shorter name.

    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 n → ∞ a n = 0 , then ∑ n = 1 ∞ a n may or may not converge. In other words, if lim n → ∞ a n = 0 , the test is inconclusive.

    • The harmonic series is a classic example of a divergent series whose terms limit to zero. The more general class of p-series,

    ∑ n = 1 ∞ 1 n p ,

    exemplifies the possible results of the test:

  • If p ≤ 0, then the term test identifies the series as divergent.
  • If 0 < p ≤ 1, then the term test is inconclusive, but the series is divergent by the integral test for convergence.
  • If 1 < p, then the term test is inconclusive, but the series is convergent, again by the integral test for convergence.

    • Proofs

    The test is typically proved in contrapositive form:

  • If ∑ n = 1 ∞ a n converges, then lim n → ∞ a n = 0.

    • Limit manipulation

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

    lim n → ∞ s n = L

    for some number s. Then

    lim n → ∞ a n = lim n → ∞ ( s n − s n − 1 ) = lim n → ∞ s n − lim n → ∞ s n − 1 = L − L = 0.

    Cauchy's criterion

    The assumption that the series converges means that it passes Cauchy's convergence test: for every ε > 0 there is a number N such that

    | a n + 1 + a n + 2 + … + a n + p | < ε

    holds for all n > N and p ≥ 1. Setting p = 1 recovers the definition of the statement

    lim n → ∞ 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 space (or any (additively written) abelian group).

    References

    Term test Wikipedia
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