Convergence (mathematics)
   HOME



Convergence (mathematics)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ... is the sum of the terms of an infinite sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ... of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series Series may refer to: People with the name * Caroline Series (born 195 ...
[...More Info...]      
[...Related Items...]



picture info

Mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ... and number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...), formulas and related structures (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...), shapes and spaces in which they are contained (geometry Geometry (from the grc, ...
[...More Info...]      
[...Related Items...]



picture info

E (mathematical Constant)
The number , also known as Euler's number, is a mathematical constant A mathematical constant is a key whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an ), or by mathematicians' names to facilitate using it across multiple s. Constants arise in many areas of , with constan ... approximately equal to 2.71828, and can be characterized in many ways. It is the base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ... of the natural logarithm The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura .... It is the limit Limit or Limits may refer to: ...
[...More Info...]      
[...Related Items...]



picture info

Integral Test For Convergence
In mathematics, the integral test for convergence is a convergence tests, method used to test infinite series (mathematics), series of Monotonic function, monotonous terms for convergent series, convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Statement of the test Consider an integer and a function defined on the unbounded interval (mathematics), interval , on which it is monotone decreasing. Then the infinite series :\sum_^\infty f(n) converges to a real number if and only if the improper integral :\int_N^\infty f(x)\,dx is finite. In particular, if the integral diverges, then the divergent series, series diverges as well. Remark If the improper integral is finite, then the proof also gives the upper and lower bounds, lower and upper bounds for the infinite series. Note that if the function f is increasing, then the function -f is decreasing and the above theorem applies. Proof The proof ...
[...More Info...]      
[...Related Items...]



picture info

Limit Superior
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the limit inferior and limit superior of a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ... (see limit of a func ...
[...More Info...]      
[...Related Items...]



picture info

Non-negative
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the sign of a real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ... is its property of being either positive, negative, or zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and .... Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third si ...
[...More Info...]      
[...Related Items...]



picture info

Root Test
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the root test is a criterion for the convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ... (a convergence test In mathematics, convergence tests are methods of testing for the Convergent series, convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an series (mathematics), infinite series \sum_^\infty a_n. Lis ...) of an infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory) ...
[...More Info...]      
[...Related Items...]



picture info

Ratio Test
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the ratio test is a test Test(s), testing, or TEST may refer to: * Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities Arts and entertainment * Test (2013 film), ''Test'' (2013 film), an American film * Test ( ... (or "criterion") for the convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ... of a series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (19 ...
[...More Info...]      
[...Related Items...]



Direct Comparison Test
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...), provides a way of deducing the convergence or divergence of an infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysi ...
[...More Info...]      
[...Related Items...]



Comparison Test Series
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are Similarity (psychology), similar to the other, which are Difference (philosophy), different, and to what degree. Where characteristics are different, the differences may then be evaluated to determine which thing is best suited for a particular purpose. The description of similarities and differences found between the two things is also called a comparison. Comparison can take many distinct forms, varying by field: To compare things, they must have characteristics that are similar enough in relevant ways to merit comparison. If two things are too different to compare in a useful way, an attempt to compare them is colloquially referred to in English as "comparing apples and oranges." Comparison is widely used in society, in science and in the arts. General usage Comparison is a natural ...
[...More Info...]      
[...Related Items...]



Reciprocal Fibonacci Constant
The reciprocal Fibonacci constant, or ψ, is defined as the sum of the reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...s of the Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), ...s: :\psi = \sum_^ \frac = \frac + \frac + \frac + \frac + \frac + \frac + \frac + \frac + \cdots. The ratio of successive terms in this sum tends to the reciprocal of the golden ratio In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), ...
[...More Info...]      
[...Related Items...]



picture info

Fibonacci Number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ..., the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. Starting from 0 and 1, the next few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ..., as ea ...
[...More Info...]      
[...Related Items...]



picture info

Geometric Series
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a geometric series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ... is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as ''a'' + ''ar'' + ''ar''2 + ''ar''3 + ... , where ''a'' is the coefficient In mathematics Mathematics (from Greek: ) i ...
[...More Info...]      
[...Related Items...]