“Term test”的意思、由来-开放百科全书

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Term test

释义

Usage
Proofs
Limit manipulation Cauchy's criterion
Scope
Notes
References

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

If or if the limit does not exist, then diverges.

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

When testing if a series converges or diverges, this test is often checked first due to its ease of use.

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 then may or may not converge. In other words, if the test is inconclusive.

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:

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 converges, then

Limit manipulation

If s_n 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]

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^[6] (or any (additively written) abelian group).

Notes

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

References

{{cite book |last=Brabenec |first=Robert |title=Resources for the study of real analysis |year=2005 |publisher=MAA |isbn=0883857375}}
{{cite book |last=Hansen |first=Vagn Lundsgaard |title=Functional Analysis: Entering Hilbert Space |year=2006 |publisher=World Scientific |isbn=9812565639}}
{{cite book |author=Kaczor, Wiesława and Maria Nowak |title=Problems in Mathematical Analysis |year=2003 |publisher=American Mathematical Society |isbn=0821820508}}
{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
{{cite book |last=Șuhubi |first=Erdoğan S. |title=Functional Analysis |year=2003 |publisher=Springer |isbn=1402016166}}

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