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series (mathematics)
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{{about|infinite sums|finite sums|Summation}}{{Calculus}}In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.BOOK, Calculus Made Easy, Thompson, Silvanus, Silvanus P. Thompson, Gardner, Martin, Martin Gardner, 1998, 978-0-312-18548-0, The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 19th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.In modern terminology, any (ordered) infinite sequence (a_1,a_2,a_3,ldots) of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the a_i one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like
a_1+a_2+a_3+cdots,
or, using the summation sign,
sum_{i=1}^infty a_i.
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as {{math|n}} tends to infinity (if the limit exists) of the finite sums of the {{math|n}} first terms of the series, which are called the {{math|n}}th partial sums of the series. That is,
sum_{i=1}^infty a_i = lim_{ntoinfty} sum_{i=1}^n a_i.
When this limit exists, one says that the series is convergent or summable, or that the sequence (a_1,a_2,a_3,ldots) is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.Generally, the terms of a series come from a ring, often the field mathbb R of the real numbers or the field mathbb C of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.

Basic properties

An infinite series or simply a series is an infinite sum, represented by an infinite expression of the form{{harvnb|Swokowski|1983|loc=p. 501}}
a_0 + a_1 + a_2 + cdots,
where (a_n) is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms a_0,a_1,dots by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as
sum_{n=0}^{infty}a_n .
If an abelian group A of terms has a concept of limit (for example, if it is a metric space), then some series, the convergent series, can be interpreted as having a value in A, called the sum of the series. This includes the common cases from calculus in which the group is the field of real numbers or the field of complex numbers. Given a series s=sum_{n=0}^infty a_n, its kth partial sum is
s_k = sum_{n=0}^{k}a_n = a_0 + a_1 + cdots + a_k.
By definition, the series sum_{n=0}^{infty} a_n converges to the limit {{math|L}} (or simply sums to {{math|L}}), if the sequence of its partial sums has a limit {{math|L}}. In this case, one usually writes
L = sum_{n=0}^{infty}a_n.
A series is said to be convergent if it converges to some limit or divergent when it does not. The value of this limit, if it exists, is then the value of the series.

Convergent series

File:Geometric sequences.svg|thumb|right|Illustration of 3 geometric seriesgeometric seriesA series {{math|∑a'n}} is said to converge or to be convergent when the sequence {{math|(s'k)}} of partial sums has a finite limit. If the limit of {{math|sk}} is infinite or does not exist, the series is said to diverge.{{citation|title=Calculus|author=Michael Spivak}} When the limit of partial sums exists, it is called the value (or sum) of the series
sum_{n=0}^infty a_n = lim_{ktoinfty} s_k = lim_{ktoinfty} sum_{n=0}^k a_n.
An easy way that an infinite series can converge is if all the an are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.Working out the properties of the series that converge even if infinitely many terms are non-zero is the essence of the study of series. Consider the example
1 + frac{1}{2}+ frac{1}{4}+ frac{1}{8}+cdots+ frac{1}{2^n}+cdots.
It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next Â¼. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted {{math|S}}, it can be seen that
S/2 = frac{1+ frac{1}{2}+ frac{1}{4}+ frac{1}{8}+cdots}{2} = frac{1}{2}+ frac{1}{4}+ frac{1}{8}+ frac{1}{16} +cdots.
Therefore,
S-S/2 = 1 Rightarrow S = 2.
The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in
x = 0.111dots ,
encodes the series
sum_{n=1}^infty frac{1}{10^n}.
Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that {{nowrap|1=9 × 0.111... = 0.999... = 1}}, which only relies on the fact that the limit laws for series preserve the arithmetic operations; this argument is presented in the article 0.999....

Examples of numerical series

{{For|other examples|List of mathematical series|Sums of reciprocals#Infinitely many terms}}
  • A geometric series is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). Example:


1 + {1 over 2} + {1 over 4} + {1 over 8} + {1 over 16} + cdots=sum_{n=0}^infty{1 over 2^n}.
In general, the geometric series
sum_{n=0}^infty z^n
converges if and only if |z| < 1.


3 + {5 over 2} + {7 over 4} + {9 over 8} + {11 over 16} + cdots=sum_{n=0}^infty{(3+2n) over 2^n}.


1 + {1 over 2} + {1 over 3} + {1 over 4} + {1 over 5} + cdots = sum_{n=1}^infty {1 over n}.
The harmonic series is divergent.


1 - {1 over 2} + {1 over 3} - {1 over 4} + {1 over 5} - cdots =sum_{n=1}^infty {left(-1right)^{n-1} over n}=ln(2) quad (alternating harmonic series)
and
-1+frac{1}{3} - frac{1}{5} + frac{1}{7} - frac{1}{9} + cdots =sum_{n=1}^infty frac{left(-1right)^n}{2n-1} = -frac{pi}{4}


sum_{n=1}^inftyfrac{1}{n^p}
converges if p > 1 and diverges for p ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of p, the sum of this series is Riemann's zeta function.


sum_{n=1}^infty (b_n-b_{n+1})
converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 − L.
  • There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series {{anchor|Flint Hills series|Flint Hills Series}}


sum_{n=1}^infty frac{csc^{2} n}{n^{3}}
converges or not. The convergence depends on how well pi can be approximated with rational numbers (which is unknown as of yet). More specifically, the values of n with large numerical contributions to the sum are the numerators of the continued fraction convergents of pi, a sequence beginning with 1, 3, 22, 333, 355, 103993, ... {{OEIS|A046947}}. These are integers that are close to npi for some integer n, so that sin npi is close to 0 and its reciprocal is large. Alekseyev (2011) proved that if the series converges, then the irrationality measure of pi is smaller than 2.5, which is much smaller than the current known bound of 7.6063....Max A. Alekseyev, On convergence of the Flint Hills series, arXiv:1104.5100, 2011.{{MathWorld|FlintHillsSeries|Flint Hills Series}}

Ï€

sum_{i=1}^{infty} frac{1}{i^2} = frac{1}{1^2} + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdots = frac{pi^2}{6}
sum_{i=1}^infty frac{(-1)^{i+1}(4)}{2i-1} = frac{4}{1} - frac{4}{3} + frac{4}{5} - frac{4}{7} + frac{4}{9} - frac{4}{11} + frac{4}{13} - cdots = pi

Natural logarithm of 2

sum_{i=1}^infty frac{(-1)^{i+1}}{i} = ln 2
sum_{i=0}^infty frac{1}{(2i+1)(2i+2)} = ln 2
sum_{i=0}^infty frac{(-1)^i}{(i+1)(i+2)} = 2ln(2) -1
sum_{i=1}^infty frac{1}{i(4i^2-1)} = 2ln(2) -1
sum_{i=1}^infty frac{1}{2^{i}i} = ln 2
sum_{i=1}^infty left(frac{1}{3^i}+frac{1}{4^i}right)frac{1}{i} = ln 2
sum_{i=1}^infty frac{1}{2i(2i-1)} = ln 2

Natural logarithm base e

sum_{i = 0}^infty frac{(-1)^i}{i!} = 1-frac{1}{1!}+frac{1}{2!}-frac{1}{3!}+cdots = frac{1}{e}
sum_{i = 0}^infty frac{1}{i!} = frac{1}{0!} + frac{1}{1!} + frac{1}{2!} + frac{1}{3!} + frac{1}{4!} + cdots = e

Calculus and partial summation as an operation on sequences

Partial summation takes as input a sequence, { a'n }, and gives as output another sequence, { S'N }. It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, Δ. These behave as discrete analogs of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence {1, 1, 1, ...} has series {1, 2, 3, 4, ...} as its partial summation, which is analogous to the fact that int_0^x 1,dt = x.In computer science it is known as prefix sum.

Properties of series

Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.

Non-negative terms

When an is a non-negative real number for every n, the sequence SN of partial sums is non-decreasing. It follows that a series ∑an with non-negative terms converges if and only if the sequence SN of partial sums is bounded.For example, the series
sum_{n = 1}^infty frac{1}{n^2}
is convergent, because the inequality
frac1 {n^2} le frac{1}{n-1} - frac{1}{n}, quad n ge 2,
and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the Basel problem.

Absolute convergence

A series
sum_{n=0}^infty a_n
is said to converge absolutely if the series of absolute values
sum_{n=0}^infty left|a_nright|
converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.

Conditional convergence

A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series
sumlimits_{n=1}^infty {(-1)^{n+1} over n} = 1 - {1 over 2} + {1 over 3} - {1 over 4} + {1 over 5} - cdots
which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the an are real and S is any real number, that one can find a reordering so that the reordered series converges with sum equal to S.Abel's test is an important tool for handling semi-convergent series. If a series has the form
sum a_n = sum lambda_n b_n
where the partial sums B'N = {{nowrap|b0 + ··· + bn}} are bounded, λ'n has bounded variation, and {{nowrap|lim λnBn}} exists:
sup_N Bigl| sum_{n=0}^N b_n Bigr| < infty, sum |lambda_{n+1} - lambda_n| < infty text{and} lambda_n B_n text{converges,}
then the series {{nowrap|∑ an}} is convergent. This applies to the pointwise convergence of many trigonometric series, as in
sum_{n=2}^infty frac{sin(n x)}{ln n}
with 0 < x < 2Ï€. Abel's method consists in writing b'n+1 = B'n+1 âˆ’ Bn, and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series {{nowrap|∑ an}} to the absolutely convergent series
sum (lambda_n - lambda_{n+1}) , B_n.

Convergence tests

There exist many tests that can be used to determine whether particular series converge or diverge.
  • n-th term test: If limn→∞ a'n â‰  0, then the series diverges; if limn→∞ a'n = 0, then the test is inconclusive.
  • Comparison test 1 (see Direct comparison test): If ∑bn  is an absolutely convergent series such that |an | â‰¤ C |bn | for some number C  and for sufficiently large n , then ∑an  converges absolutely as well. If ∑|bn | diverges, and |an | â‰¥ |bn | for all sufficiently large n , then ∑an  also fails to converge absolutely (though it could still be conditionally convergent, for example, if the an  alternate in sign).
  • Comparison test 2 (see Limit comparison test): If ∑bn  is an absolutely convergent series such that |an+1 /an | â‰¤ |bn+1 /bn | for sufficiently large n , then ∑an  converges absolutely as well. If ∑|bn | diverges, and |an+1 /an | â‰¥ |bn+1 /bn | for all sufficiently large n , then ∑an  also fails to converge absolutely (though it could still be conditionally convergent, for example, if the an  alternate in sign).
  • Ratio test: If there exists a constant C < 1 such that |a'n+1/a'n|, the series


sum_{n=0}^infty varepsilon_n a_n
converges in X. If X  is a Banach space, then one may define the notion of absolute convergence. A series ∑an of vectors in X  converges absolutely if
sum_{n in mathbf{N}} |a_n| < +infty.
If a series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of {{harvtxt|Dvoretzky|Rogers|1950}}).

Well-ordered sums

Conditionally convergent series can be considered if I is a well-ordered set, for example, an ordinal number α0. One may define by transfinite recursion:
sum_{beta < alpha + 1} a_beta = a_{alpha} + sum_{beta < alpha} a_beta
and for a limit ordinal α,
sum_{beta < alpha} a_beta = lim_{gammatoalpha} sum_{beta < gamma} a_beta
if this limit exists. If all limits exist up to α0, then the series converges.

Examples

{{ordered list|1= Given a function f : X→Y, with Y an abelian topological group, define for every a âˆˆ X
f_a(x)=
begin{cases}f(a) & x=a, end{cases}a function whose support is a singleton {a}. Then
f=sum_{a in X}f_a
in the topology of pointwise convergence (that is, the sum is taken in the infinite product group YX ).|2= In the definition of partitions of unity, one constructs sums of functions over arbitrary index set I,
sum_{i in I} varphi_i(x) = 1.
While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is locally finite, that is, for every x there is a neighborhood of x in which all but a finite number of functions vanish. Any regularity property of the φi,  such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.|3= On the first uncountable ordinal ω1 viewed as a topological space in the order topology, the constant function f: [0,ω1) → [0,ω1] given by f(α) = 1 satisfies
sum_{alphain[0,omega_1)}f(alpha) = omega_1
(in other words, ω1 copies of 1 is ω1) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.}}

See also

Notes

{{reflist}}

References

  • Bromwich, T. J. An Introduction to the Theory of Infinite Series MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.
  • JOURNAL, 10.1073/pnas.36.3.192, Dvoretzky, Aryeh, Rogers, C. Ambrose, Absolute and unconditional convergence in normed linear spaces, Proc. Natl. Acad. Sci. U.S.A., 36, 3, 1950, 192–197, 1063182, 1950PNAS...36..192D
,
  • {{citation|first=Earl W.|last=Swokowski|title=Calculus with analytic geometry|edition=Alternate|year=1983|publisher=Prindle, Weber & Schmidt|place=Boston|isbn=978-0-87150-341-1}}
{{MathSciNet|id=0033975}}

External links

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