Term Test
In mathematics, the ''n''th-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:If \lim_ a_n \neq 0 or if the limit does not exist, then \sum_^\infty a_n diverges.Many authors do not name this test or give it a shorter name.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. 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 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 \lim_ a_n = 0, then \sum_^\infty a_n may or may not converge. In other words, if \lim_ a_ ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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