Nth-term test

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

• If $\lim _{n\to \infty }a_{n}\neq 0$ or if the limit does not exist, then $\sum _{n=1}^{\infty }a_{n}$ diverges.

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

Usage

well nth term is a sequence of numbers that i struggle with Mr Magill!!!.

Proofs

The test is typically proved in contrapositive form:

• If $\sum _{n=1}^{\infty }a_{n}$ converges, then $\lim _{n\to \infty }a_{n}=0.$

Limit manipulation

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

$\lim _{n\to \infty }s_{n}=s$

for some number s. Then

$\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }(s_{n}-s_{n-1})=s-s=0.$

Cauchy's criterion

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

$|a_{n+1}+a_{n+2}+\ldots +a_{n+p}|<\varepsilon$

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

$\lim _{n\to \infty }a_{n}=0.$

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.

References

• Brabenec, Robert (2005). Resources for the study of real analysis. MAA. ISBN 0883857375.
• Hansen, Vagn Lundsgaard (2006). Functional Analysis: Entering Hilbert Space. World Scientific. ISBN 9812565639.
• Kaczor, Wiesława and Maria Nowak (2003). Problems in Mathematical Analysis. American Mathematical Society. ISBN 0821820508.
• Rudin, Walter (1976) [1953]. Principles of mathematical analysis (3e ed.). McGraw-Hill. ISBN 0-07-054235-X.
• Stewart, James (1999). Calculus: Early transcendentals (4e ed.). Brooks/Cole. ISBN 0-534-36298-2.
• Șuhubi, Erdoğan S. (2003). Functional Analysis. Springer. ISBN 1402016166.