NationMaster - Encyclopedia: Convergent series

In mathematics, a series is the sum of the terms of a sequence of numbers. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Summation is the addition of a set of numbers; the result is their sum. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...

Given a sequence $left { a_1, a_2, a_3,... right }$ , the nth partial sum $S n$ is the sum of the first n terms of the sequence, that is, In mathematics, a series is a sum of a sequence of terms. ...

$S_N = sum_{n=1}^N a_n$ .

A series is convergent if the sequence of its partial sums $left { S_1, S_2, S_3,... right }$ converges. In more formal language, a series converges if there exists a limit $ell$ such that for any arbitrarily small positive number $ε > 0$ , there is a large integer $N$ such that for all $n ge N$ , In mathematics, a series is often represented as the sum of a sequence of terms. ... Limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... The integers are commonly denoted by the above symbol. ...

$left | S_n - ell right vert le epsilon$ .

1 Examples of convergent and divergent series
2 Convergence tests
3 Conditional and absolute convergence
4 Uniform convergence
5 Cauchy convergence criterion
6 References

Examples of convergent and divergent series

The reciprocals of powers of 2 produce a convergent series:

${1 over 1}+{1 over 2}+{1 over 4}+{1 over 8}+{1 over 16}+{1 over 32}+cdots = 2.$
The reciprocals of positive integers produce a divergent series:

${1 over 1}+{1 over 2}+{1 over 3}+{1 over 4}+{1 over 5}+{1 over 6}+cdots$
Alternating the signs of the reciprocals of positive integers produces a convergent series:

${1 over 1}-{1 over 2}+{1 over 3}-{1 over 4}+{1 over 5}-{1 over 6}+cdots = ln 2$
The reciprocals of prime numbers produce a divergent series:

${1 over 2}+{1 over 3}+{1 over 5}+{1 over 7}+{1 over 11}+{1 over 13}+cdots$
The reciprocals of square numbers produce a convergent series:

${1 over 1}+{1 over 4}+{1 over 9}+{1 over 16}+{1 over 25}+{1 over 36}+cdots = {pi^2 over 6}$
Alternating the signs of the reciprocals of positive odd numbers produces a convergent series:

${1 over 1}-{1 over 3}+{1 over 5}-{1 over 7}+{1 over 9}-{1 over 11}+cdots = {pi over 4}$

In mathematics, a divergent series is an infinite series that does not converge. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a square number, sometimes also called a perfect square, is a positive integer that can be written as the square of some other integer. ...

Convergence tests

Main article: Convergence tests

There are a number of methods of determining whether a series converges or diverges. In mathematics, convergence tests are methods, how to determinate if a series converges or diverges. ... In mathematics, a divergent series is an infinite series that does not converge. ...

If the blue series,

Σ b n

, can be proven to converge, then the smaller series,

Σ a n

must converge. By contraposition, if the red series,

Σ a n

is proven to diverge, then

Σ b n

must also diverge.

Comparison test. The terms of the sequence $left { a_n right }$ are compared to those of another sequence $left { b_n right }$ . If, for all n, Image File history File links Comparison_test_series. ... Image File history File links Comparison_test_series. ... In mathematics, the Comparison test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ...

$0 le a_n le b_n$ , and $sum_{n=1}^infty b_n$ converges, then so does $sum_{n=1}^infty a_n$ .

However, if, for all n,

$0 le b_n le a_n$ , and $sum_{n=1}^infty b_n$ diverges, then so does $sum_{n=1}^infty a_n$ .

Ratio test. Assume that for all n, $a n > 0$ . Suppose that there exists $r$ such that In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ...

$lim_{n to infty} left|frac{a_{n+1}}{a_n}right| = r$ .

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Root test or nth root test. Suppose that the terms of the sequence in question are non-negative, and that there exists r such that In mathematics, the root test is a test for the convergence of an infinite series. ... A negative number is a number that is less than zero, such as −3. ...

$lim_{n to infty} sqrt[n]{a_n} = r$

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

Root test is equivalent to ratio test. In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...

Integral test. The series can be compared to an integral to establish convergence or divergence. Let $f (n) = a n$ be a positive and monotone decreasing function. If The integral test for convergence is a method used to test infinite series of nonnegative terms for convergence. ... In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ...

$int_{1}^{infty} f(x), dx = lim_{t to infty} int_{1}^{t} f(x), dx < infty,$

then the series converges. But if the integral diverges, then the series does so as well.

Limit comparison test. If $left { a_n right }, left { b_n right } > 0$ , and the limit $lim_{n to infty} frac{a_n}{b_n}$ exists and is not zero, then $sum_{n=1}^infty a_n$ converges if and only if $sum_{n=1}^infty b_n$ converges. It has been suggested that this article or section be merged with Logical biconditional. ...

Alternating series test. Also known as the Leibniz criterion, the alternating series test states that for an alternating series of the form $sum_{n=1}^infty a_n (-1)^n$ , if $left { a_n right }$ is monotone decreasing, and has a limit of 0, then the series converges. The alternating series test is a method used to test infinite series of terms for convergence. ... The alternating series test is a method used to test infinite series of terms for convergence. ... In mathematics, an alternating series is an infinite series of the form with an â‰¥ 0. ... This article or section does not cite its references or sources. ...

Cauchy condensation test. If $left { a_n right }$ is a monotone decreasing sequence, then $sum_{n=1}^infty a_n$ converges if and only if $sum_{k=1}^infty 2^k a_{2^k}$ converges. In mathematics, the Cauchy condensation test is a standard convergence test for infinite series. ...

Dirichlet's test In mathematics, Dirichlets test is a method of testing for the convergence of a series and is named after mathematician Johann Dirichlet. ...

Abel's test Abels Test: Relates to Dr Abel who did a substantial amount of work into chemotherapeutic regiemes. ...

Raabe's test In mathematics, the ratio test is a test (or criterion) for the convergence of a series whose terms are real or complex numbers. ...

Conditional and absolute convergence

For any sequence $left { a_1, a_2, a_3... right }$ , $a_n le left | a_n right vert$ for all n. Therefore,

$sum_{n=1}^infty a_n le sum_{n=1}^infty left | a_n right vert.$

This means that if $sum_{n=1}^infty left | a_n right vert$ converges, then $sum_{n=1}^infty a_n$ also converges (but not vice-versa).

If the series $sum_{n=1}^infty left | a_n right vert$ converges, then the series $sum_{n=1}^infty a_n$ is absolutely convergent. In mathematics, a series is a sum of a sequence of terms. ...

If the series $sum_{n=1}^infty a_n$ converges but the series $sum_{n=1}^infty left | a_n right vert$ diverges, then the series $sum_{n=1}^infty a_n$ is conditionally convergent. In mathematics, a series is a sum of a sequence of terms. ...

The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges. In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges. ...

Uniform convergence

Main article: uniform convergence.

Let $left { f_1, f_2, f_3... right }$ be a sequence of functions. The series $sum_{n=1}^infty f_n$ is said to converge uniformly to f if the sequence ${s n}$ of partial sums defined by In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ...

$s_n(x) = sum_{k=1}^infty f_k (x)$

converges uniformly to f.

There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test. In mathematics, the Weierstrass M-test is an analogue of the comparison test for infinite series, and applies to a series whose terms are themselves functions of a real variable. ...

Cauchy convergence criterion

The Cauchy convergence criterion states that a series The Cauchy convergence test is a method used to test infinite series for convergence. ...

$sum_{n=1}^infty a_n$

converges if and only if the sequence of partial sums is a Cauchy sequence. This means that for every $varepsilon > 0,$ there is a positive integer $N$ such that for all $n geq m geq N$ we have It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...

$left| sum_{k=m}^n a_k right| < varepsilon,$

which is equivalent to

$lim_{n to infty atop mto infty} sum_{k=n}^{n+m} a_k = 0.$

References

Spivak, Michael (1994). Calculus (3rd ed.). Houston, Texas: Publish or Perish, Inc. ISBN 0-91-409889-6.
Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.

Category: Mathematical series

Results from FactBites:

Series - LoveToKnow 1911 (4894 words)
	The series i - i + I - I+..., where S n is unity or zero, according as n is odd or even, is an example of an oscillating series.
	In the case of a power series there is a quantity R such that the series converges if I z < R, and diverges if I z I > R. A circle described with the origin as centre and radius R is called the circle of convergence.
	The circle of convergence may be of infinite radius as in the case of the series for sin z, viz.

absolutely convergent: Definition and Much More from Answers.com (2742 words)
	Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
	The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed the concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions.
	Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents