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| {{Calculus}}
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| A '''series''' is, informally speaking, the sum of the terms of a [[sequence]]. '''Finite sequences and series''' have defined first and last terms, whereas '''infinite sequences and series''' continue indefinitely.<ref>p 264 '''[[Jan Gullberg]]:''' ''Mathematics: from the birth of numbers,'' W.W. Norton, 1997, ISBN 0-393-04002-X</ref>
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| In [[mathematics]], given an [[infinite set|infinite]] [[sequence]] of numbers { ''a''<sub>''n''</sub> }, a '''series''' is informally the result of adding all those terms together: ''a''<sub>1</sub> + ''a''<sub>2</sub> + ''a''<sub>3</sub> + · · ·. These can be written more compactly using the [[summation]] symbol ∑. An example is the famous series from [[Zeno's paradoxes#Proposed solutions|Zeno's dichotomy]] and [[1/2 + 1/4 + 1/8 + 1/16 + · · ·|its mathematical representation]]:
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| :<math>\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots.</math>
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| The terms of the series are often produced according to a certain rule, such as by a [[formula]], or by an [[algorithm]]. As there are an infinite number of terms, this notion is often called an '''infinite series'''. Unlike finite summations, infinite series need tools from [[mathematical analysis]], and specifically the notion of [[limit (mathematics)|limits]], to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.
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| ==Basic properties==
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| {{inline citations|section|date=July 2013}}
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| ===Definition===
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| For any [[sequence]] <math>\{a_n\}</math> of [[rational numbers]], [[real numbers]], [[complex numbers]], [[Function (mathematics)|functions]] thereof, etc., the associated '''series''' is defined as the ordered [[formal sum]]
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| :<math>\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots </math>.
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| The '''sequence of partial sums''' <math>\{S_k\}</math> associated to a series <math>\sum_{n=0}^\infty a_n</math> is defined for each <math>k</math> as the sum of the sequence <math>\{a_n\}</math> from <math>a_0</math> to <math>a_k</math>
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| :<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
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| By definition the series <math>\sum_{n=0}^{\infty} a_n</math> '''converges''' to a limit <math>L</math> if and only if the associated sequence of partial sums <math>\{S_k\}</math> [[Limit_of_a_sequence#Formal_Definition|converges]] to <math>L</math>. This definition is usually written as
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| :<math>L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
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| More generally, if <math>I \xrightarrow{a} G</math> is a [[Function (mathematics)|function]] from an [[index set]] I to a set G, then the '''series''' associated to <math>a</math> is the [[formal sum]] of the elements <math>a(x) \in G </math> over the index elements <math>x \in I</math> denoted by the
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| :<math>\sum_{x \in I} a(x).</math>
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| When the index set is the natural numbers <math>I=\mathbb{N}</math>, the function <math>\mathbb{N} \xrightarrow{a} G</math> is a [[sequence]] denoted by <math>a(n)=a_n</math>. A series indexed on the natural numbers is an ordered formal sum and so we rewrite <math>\sum_{n \in \mathbb{N}}</math> as <math>\sum_{n=0}^{\infty}</math> in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
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| :<math>\sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots.</math>
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| When the set <math>G</math> is a [[semigroup]], the '''sequence of partial sums''' <math>\{S_k\} \subset G</math> associated to a sequence <math>\{a_n\} \subset G</math> is defined for each <math>k</math> as the sum of the terms <math>a_0,a_1,\cdots,a_k</math>
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| :<math>S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k.</math>
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| When the semigroup <math>G</math> is also a [[topological space]], then the series <math>\sum_{n=0}^{\infty} a_n</math> '''converges''' to an element <math>L \in G</math> if and only if the associated sequence of partial sums <math>\{S_k\}</math> [[Limit_of_a_sequence#Formal_Definition|converges]] to <math>L</math>. This definition is usually written as
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| :<math>L = \sum_{n=0}^{\infty} a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k.</math>
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| ===Convergent series===
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| A series  ∑''a<sub>n</sub>''  is said to ''''[[Convergent series|converge]]'''' or to 'be convergent' when the sequence ''S''<sub>''N''</sub> of partial sums has a finite [[Limit of a sequence|limit]]. If the limit of ''S''<sub>''N''</sub> is infinite or does not exist, the series is said to '''[[Divergent series|diverge]]'''. When the limit of partial sums exists, it is called the '''sum of the series'''
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| : <math>\sum_{n=0}^\infty a_n = \lim_{N\to\infty} S_N = \lim_{N\to\infty} \sum_{n=0}^N a_n.</math>
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| An easy way that an infinite series can converge is if all the ''a''<sub>''n''</sub> 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.
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| 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
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| :<math> 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots.</math>
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| It is possible to "visualize" its convergence on the [[real number|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 ¼. 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 ''S'', it can be seen that
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| :<math>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.</math>
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| Therefore,
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| :<math>S-S/2 = 1 \Rightarrow S = 2.\,\!</math>
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| Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For instance, when we talk about a [[Repeating decimal|recurring decimal]], as in
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| :<math>x = 0.111\dots \, </math>
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| we are talking, in fact, just about the series
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| :<math>\sum_{n=1}^\infty \frac{1}{10^n}.</math>
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| But since these series always converge to [[real numbers]] (because of what is called the [[Complete space|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, it should offend no sensibilities if we make no distinction between 0.111… and <sup>1</sup>/<sub>9</sub>. Less clear is the argument that {{nowrap|1=9 × 0.111… = 0.999… = 1}}, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See [[0.999...]] for more.
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| ===Examples===
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| * A ''[[geometric series]]'' is one where each successive term is produced by multiplying the previous term by a [[Mathematical constant|constant number]] (called the common ratio in this context). Example:
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| ::<math>1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n}.</math>
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| :In general, the geometric series
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| ::<math>\sum_{n=0}^\infty z^n</math>
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| :converges [[if and only if]] |''z''| < 1.
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| * An ''[[Arithmetico-geometric sequence]]'' is a generalization of the geometric series, which has coefficients of the common ratio equal to the terms in an [[arithmetic series]]. Example:
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| ::<math>3 + {5 \over 2} + {7 \over 4} + {9 \over 8} + {11 \over 16} + \cdots=\sum_{n=0}^\infty{(3+2n) \over 2^n}.</math>
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| * The ''[[harmonic series (mathematics)|harmonic series]]'' is the series
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| ::<math>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots = \sum_{n=1}^\infty {1 \over n}.</math>
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| :The harmonic series is [[harmonic series (mathematics)#Divergence|divergent]].
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| * An ''[[alternating series]]'' is a series where terms alternate signs. Example:
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| ::<math>1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum_{n=1}^\infty (-1)^{n+1} {1 \over n}=\ln(2).</math>
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| *The [[harmonic series (mathematics)#P-series|p-series]]
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| ::<math>\sum_{n=1}^\infty\frac{1}{n^r}</math>
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| :converges if ''r'' > 1 and diverges for ''r'' ≤ 1, which can be shown with the integral criterion described below in [[Series (mathematics)#Convergence tests|convergence tests]]. As a function of ''r'', the sum of this series is [[Riemann zeta function|Riemann's zeta function]].
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| *A [[telescoping series]]
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| ::<math>\sum_{n=1}^\infty (b_n-b_{n+1})</math>
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| :converges if the [[sequence]] ''b''<sub>''n''</sub> converges to a limit ''L'' as ''n'' goes to infinity. The value of the series is then ''b''<sub>1</sub> − ''L''.
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| ===Calculus and partial summation as an operation on sequences===
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| Partial summation takes as input a sequence, { ''a''<sub>''n''</sub> }, and gives as output another sequence, { ''S''<sub>''N''</sub> }. It is thus a [[unary operation]] on sequences. Further, this function is [[linear function|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 [[integral|integration]] and [[derivative|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 <math>\int_0^x 1\,dt = x.</math>
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| In [[computer science]] it is known as [[prefix sum]].
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| ==Properties of series==
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| Series are classified not only by whether they converge or diverge, but also by the properties of the terms a<sub>n</sub> (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a<sub>n</sub> (whether it is a real number, arithmetic progression, trigonometric function); etc.
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| ===Non-negative terms===
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| When ''a<sub>n</sub>'' is a non-negative real number for every ''n'', the sequence ''S<sub>N</sub>'' of partial sums is non-decreasing. It follows that a series ∑''a<sub>n</sub>'' with non-negative terms converges if and only if the sequence ''S<sub>N</sub>'' of partial sums is bounded.
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| For example, the series
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| :<math>\sum_{n \ge 1} \frac{1}{n^2}</math>
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| is convergent, because the inequality
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| :<math>\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, \quad n \ge 2,</math>
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| and a telescopic sum argument implies that the partial sums are bounded by 2.
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| ===Absolute convergence===
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| {{Main|Absolute convergence}}
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| A series
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| :<math>\sum_{n=0}^\infty a_n</math>
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| is said to '''converge absolutely''' if the series of [[absolute value]]s
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| :<math>\sum_{n=0}^\infty \left|a_n\right|</math>
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| 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.
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| ===Conditional convergence===
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| {{Main|Conditional convergence}}
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| 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
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| :<math>\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots</math>
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| 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 (mathematics)|harmonic series]]. The [[Riemann series theorem]] says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the ''a''<sub>''n''</sub> 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''.
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| [[Abel's test]] is an important tool for handling semi-convergent series. If a series has the form
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| :<math>\sum a_n = \sum \lambda_n b_n</math>
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| where the partial sums ''B''<sub>''N''</sub> = {{nowrap|''b''<sub>0</sub> + ··· + ''b<sub>n</sub>''}} are bounded, ''λ''<sub>''n''</sub> has bounded variation, and {{nowrap|lim λ<sub>''n''</sub> ''B''<sub>''n''</sub>}} exists:
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| :<math>\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,}</math>
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| then the series {{nowrap|∑ ''a<sub>n</sub>''}} is convergent. This applies to the pointwise convergence of many trigonometric series, as in
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| :<math>\sum_{n=2}^\infty \frac{\sin(n x)}{\ln n}</math>
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| with 0 < ''x'' < 2π. Abel's method consists in writing ''b''<sub>''n''+1</sub> = ''B''<sub>''n''+1</sub> − ''B''<sub>''n''</sub>, and in performing a transformation similar to [[integration by parts]] (called [[summation by parts]]), that relates the given series {{nowrap|∑ ''a<sub>n</sub>''}} to the absolutely convergent series
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| :<math> \sum (\lambda_n - \lambda_{n+1}) \, B_n.</math>
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| ==Convergence tests==
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| {{Main|Convergence tests}}
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| * ''[[n-th term test]]'': If lim<sub>''n''→∞</sup> ''a''<sub>''n''</sub> ≠ 0 then the series diverges.
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| *Comparison test 1 (see [[Direct comparison test]]): If ∑''b<sub>n</sub>'' is an [[absolute convergence|absolutely convergent]] series such that |''a<sub>n</sub>'' | ≤ ''C'' |''b<sub>n</sub>'' | for some number ''C''  and for sufficiently large ''n'' , then ∑''a<sub>n</sub>''  converges absolutely as well. If ∑|''b<sub>n</sub>'' | diverges, and |''a<sub>n</sub>'' | ≥ |''b<sub>n</sub>'' | for all sufficiently large ''n'' , then ∑''a<sub>n</sub>''  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a<sub>n</sub>'' alternate in sign).
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| *Comparison test 2 (see [[Limit comparison test]]): If ∑''b<sub>n</sub>''  is an absolutely convergent series such that |''a<sub>n+1</sub>'' /''a<sub>n</sub>'' | ≤ |''b<sub>n+1</sub>'' /''b<sub>n</sub>'' | for sufficiently large ''n'' , then ∑''a<sub>n</sub>''  converges absolutely as well. If ∑|''b<sub>n</sub>'' | diverges, and |''a<sub>n+1</sub>'' /''a<sub>n</sub>'' | ≥ |''b<sub>n+1</sub>'' /''b<sub>n</sub>'' | for all sufficiently large ''n'' , then ∑''a<sub>n</sub>''  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the ''a<sub>n</sub>''  alternate in sign).
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| *[[Ratio test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''+1</sub>/''a''<sub>''n''</sub>|<''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely. When the ratio is less than 1, but not less than a constant less than 1, convergence is possible but this test does not establish it.
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| *[[Root test]]: If there exists a constant ''C'' < 1 such that |''a''<sub>''n''</sub>|<sup>1/''n''</sup> ≤ ''C'' for all sufficiently large ''n'', then ∑''a''<sub>''n''</sub> converges absolutely.
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| *[[Integral test for convergence|Integral test]]: if ''ƒ''(''x'') is a positive [[monotone decreasing]] function defined on the [[interval (mathematics)|interval]] <nowiki>[</nowiki>1, ∞<nowiki>)</nowiki><!--DO NOT "FIX" THE "TYPO" IN THE FOREGOING. IT IS INTENDED TO SAY [...) WITH A SQUARE BRACKET ON THE LEFT AND A ROUND BRACKET ON THE RIGHT. --> with ''ƒ''(''n'') = ''a''<sub>''n''</sub> for all ''n'', then ∑''a''<sub>''n''</sub> converges if and only if the [[integral]]  ∫<sub>1</sub><sup>∞</sup> ''ƒ''(''x'') d''x'' is finite.
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| *[[Cauchy's condensation test]]: If ''a''<sub>''n''</sub> is non-negative and non-increasing, then the two series  ∑''a''<sub>''n''</sub>  and  ∑2<sup>''k''</sup>''a''<sub>(2<sup>''k''</sup>)</sub> are of the same nature: both convergent, or both divergent.
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| *[[Alternating series test]]: A series of the form ∑(−1)<sup>''n''</sup> ''a''<sub>''n''</sub> (with ''a''<sub>''n''</sub> ≥ 0) is called ''alternating''. Such a series converges if the [[sequence]] ''a''<sub>''n''</sub> is [[monotone decreasing]] and converges to 0. The converse is in general not true.
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| *For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]].
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| ==Series of functions==
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| {{Main|Function series}}
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| A series of real- or complex-valued functions
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| :<math>\sum_{n=0}^\infty f_n(x)</math>
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| '''[[Pointwise convergence|converges pointwise]]''' on a set ''E'', if the series converges for each ''x'' in ''E'' as an ordinary series of real or complex numbers. Equivalently, the partial sums
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| :<math>s_N(x) = \sum_{n=0}^N f_n(x)</math>
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| converge to ''ƒ''(''x'') as ''N'' → ∞ for each ''x'' ∈ ''E''.
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| A stronger notion of convergence of a series of functions is called '''[[uniform convergence]]'''. The series converges uniformly if it converges pointwise to the function ''ƒ''(''x''), and the error in approximating the limit by the ''N''th partial sum,
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| :<math>|s_N(x) - f(x)|\ </math>
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| can be made minimal ''independently'' of ''x'' by choosing a sufficiently large ''N''.
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| Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the ''ƒ''<sub>''n''</sub> are [[integral|integrable]] on a closed and bounded interval ''I'' and converge uniformly, then the series is also integrable on ''I'' and can be integrated term-by-term. Tests for uniform convergence include the [[Weierstrass M-test|Weierstrass' M-test]], [[Abel's uniform convergence test]], [[Dini's test]]<!-- , and the [[Cauchy criterion]]: this is not about convergence of functions, even less about uniform convergence. -->.
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| More sophisticated types of convergence of a series of functions can also be defined. In [[measure theory]], for instance, a series of functions converges [[almost everywhere]] if it converges pointwise except on a certain set of [[null set|measure zero]]. Other [[modes of convergence]] depend on a different [[metric space]] structure on the space of functions under consideration. For instance, a series of functions '''converges in mean''' on a set ''E'' to a limit function ''ƒ'' provided
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| :<math>\int_E \left|s_N(x)-f(x)\right|^2\,dx \to 0</math>
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| as ''N'' → ∞.
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| ===Power series===
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| :{{Main|Power series}}
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| A '''power series''' is a series of the form
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| :<math>\sum_{n=0}^\infty a_n(x-c)^n.</math>
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| The [[Taylor series]] at a point ''c'' of a function is a power series that, in many cases, converges to the function in a neighborhood of ''c''. For example, the series
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| :<math>\sum_{n=0}^{\infty} \frac{x^n}{n!}</math>
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| is the Taylor series of <math>e^x</math> at the origin and converges to it for every ''x''.
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| Unless it converges only at ''x''=''c'', such a series converges on a certain open disc of convergence centered at the point ''c'' in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the [[radius of convergence]], and can in principle be determined from the asymptotics of the coefficients ''a''<sub>''n''</sub>. The convergence is uniform on [[closed set|closed]] and [[bounded set|bounded]] (that is, [[compact set|compact]]) subsets of the interior of the disc of convergence: to wit, it is [[Compact convergence|uniformly convergent on compact sets]].
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| Historically, mathematicians such as [[Leonhard Euler]] operated liberally with infinite series, even if they were not convergent.
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| When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
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| However, the formal operation with non-convergent series has been retained in rings of [[formal power series]] which are studied in [[abstract algebra]]. Formal power series are also used in [[combinatorics]] to describe and study [[sequence]]s that are otherwise difficult to handle; this is the method of [[generating function]]s.
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| ===Laurent series===
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| {{Main|Laurent series}}
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| Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form
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| :<math>\sum_{n=-\infty}^\infty a_n x^n.</math>
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| If such a series converges, then in general it does so in an [[annulus (mathematics)|annulus]] rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
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| ===Dirichlet series===
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| :{{Main|Dirichlet series}}
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| A [[Dirichlet series]] is one of the form
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| :<math>\sum_{n=1}^\infty {a_n \over n^s},</math>
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| where ''s'' is a [[complex number]]. For example, if all ''a''<sub>''n''</sub> are equal to 1, then the Dirichlet series is the [[Riemann zeta function]]
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| :<math>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</math>
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| Like the zeta function, Dirichlet series in general play an important role in [[analytic number theory]]. Generally a Dirichlet series converges if the real part of ''s'' is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an [[analytic function]] outside the domain of convergence by [[analytic continuation]]. For example, the Dirichlet series for the zeta function converges absolutely when Re ''s'' > 1, but the zeta function can be extended to a holomorphic function defined on <math>\mathbf{C}\setminus\{1\}</math>  with a simple [[pole (complex analysis)|pole]] at 1.
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| This series can be directly generalized to [[general Dirichlet series]].
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| | |
| ===Trigonometric series===
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| {{Main|Trigonometric series}}
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| A series of functions in which the terms are [[trigonometric function]]s is called a '''trigonometric series''':
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| :<math>\tfrac12 A_0 + \sum_{n=1}^\infty \left(A_n\cos nx + B_n \sin nx\right).</math>
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| The most important example of a trigonometric series is the [[Fourier series]] of a function.
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| | |
| ==History of the theory of infinite series==
| |
| | |
| ===Development of infinite series===
| |
| [[Greek mathematics|Greek]] mathematician [[Archimedes]] produced the first known summation of an infinite series with a
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| method that is still used in the area of calculus today. He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the summation of an infinite series, and gave a remarkably accurate approximation of [[Pi|π]].<ref>{{cite web | title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F. | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|accessdate= 2007-08-07}}</ref><ref>[http://eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ502088&ERICExtSearch_SearchType_0=no&accno=EJ502088 Archimedes and Pi-Revisited.]</ref>
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| In the 17th century, [[James Gregory (astronomer and mathematician)|James Gregory]] worked in the new [[decimal]] system on infinite series and published several [[Maclaurin series]]. In 1715, a general method for constructing the [[Taylor series]] for all functions for which they exist was provided by [[Brook Taylor]]. [[Leonhard Euler]] in the 18th century, developed the theory of [[hypergeometric series]] and [[q-series]].
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| | |
| ===Convergence criteria===
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| The investigation of the validity of infinite series is considered to begin with [[Carl Friedrich Gauss|Gauss]] in the 19th century. Euler had already considered the hypergeometric series
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| :<math>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots</math>
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| on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
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| | |
| [[Cauchy]] (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by [[James Gregory (astronomer and mathematician)|Gregory]] (1668). [[Leonhard Euler]] and [[Carl Friedrich Gauss|Gauss]] had given various criteria, and [[Colin Maclaurin]] had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of [[power series]] by his expansion of a complex [[function (mathematics)|function]] in such a form.
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| [[Niels Henrik Abel|Abel]] (1826) in his memoir on the [[binomial series]]
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| :<math>1 + \frac{m}{1!}x + \frac{m(m-1)}{2!}x^2 + \cdots</math>
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| corrected certain of Cauchy's conclusions, and gave a completely
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| scientific summation of the series for complex values of <math>m</math> and <math>x</math>. He showed the necessity of considering the subject of continuity in questions of convergence.
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| Cauchy's methods led to special rather than general criteria, and
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| the same may be said of [[Joseph Ludwig Raabe|Raabe]] (1832), who made the first elaborate
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| investigation of the subject, of [[Augustus De Morgan|De Morgan]] (from 1842), whose
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| logarithmic test [[Paul du Bois-Reymond|DuBois-Reymond]] (1873) and [[Alfred Pringsheim|Pringsheim]] (1889) have
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| shown to fail within a certain region; of [[Joseph Louis François Bertrand|Bertrand]] (1842), [[Pierre Ossian Bonnet|Bonnet]]
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| (1843), [[Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration);
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| [[George Gabriel Stokes|Stokes]] (1847), [[Paucker]] (1852), [[Chebyshev]] (1852), and [[Arndt]]
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| (1853).
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| | |
| General criteria began with [[Ernst Kummer|Kummer]] (1835), and have been
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| studied by [[Gotthold Eisenstein|Eisenstein]] (1847), [[Weierstrass]] in his various
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| contributions to the theory of functions, [[Ulisse Dini|Dini]] (1867),
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| DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.
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| | |
| ===Uniform convergence===
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| The theory of [[uniform convergence]] was treated by Cauchy (1821), his
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| limitations being pointed out by Abel, but the first to attack it
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| successfully were [[Philipp Ludwig von Seidel|Seidel]] and [[George Gabriel Stokes|Stokes]] (1847–48). Cauchy took up the
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| problem again (1853), acknowledging Abel's criticism, and reaching
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| the same conclusions which Stokes had already found. Thomae used the
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| doctrine (1866), but there was great delay in recognizing the
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| importance of distinguishing between uniform and non-uniform
| |
| convergence, in spite of the demands of the theory of functions.
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| | |
| ===Semi-convergence===
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| A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not [[absolute convergence|absolutely convergent]].
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| | |
| Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834),
| |
| who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by [[Carl Johan Malmsten|Malmsten]] (1847). [[Schlömilch]] (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and [[Faulhaber's formula|Bernoulli's function]]
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| | |
| :<math>F(x) = 1^n + 2^n + \cdots + (x - 1)^n.\,</math>
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| | |
| [[Angelo Genocchi|Genocchi]] (1852) has further contributed to the theory.
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| | |
| Among the early writers was [[Josef Hoene-Wronski|Wronski]], whose "loi suprême" (1815) was hardly recognized until [[Arthur Cayley|Cayley]] (1873) brought it into
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| prominence.
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| | |
| ===Fourier series===
| |
| [[Fourier series]] were being investigated
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| as the result of physical considerations at the same time that
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| Gauss, Abel, and Cauchy were working out the theory of infinite
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| series. Series for the expansion of sines and cosines, of multiple
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| arcs in powers of the sine and cosine of the arc had been treated by
| |
| [[Jacob Bernoulli]] (1702) and his brother [[Johann Bernoulli]] (1701) and still
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| earlier by [[Franciscus Vieta|Vieta]]. Euler and [[Joseph Louis Lagrange|Lagrange]] simplified the subject,
| |
| as did [[Louis Poinsot|Poinsot]], [[Karl Schröter|Schröter]], [[James Whitbread Lee Glaisher|Glaisher]], and [[Ernst Kummer|Kummer]].
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| | |
| Fourier (1807) set for himself a different problem, to
| |
| expand a given function of ''x'' in terms of the sines or cosines of
| |
| multiples of ''x'', a problem which he embodied in his ''[[Théorie analytique de la chaleur]]'' (1822). Euler had already given the
| |
| formulas for determining the coefficients in the series;
| |
| Fourier was the first to assert and attempt to prove the general
| |
| theorem. [[Siméon Denis Poisson|Poisson]] (1820–23) also attacked the problem from a
| |
| different standpoint. Fourier did not, however, settle the question
| |
| of convergence of his series, a matter left for [[Augustin Louis Cauchy|Cauchy]] (1826) to
| |
| attempt and for Dirichlet (1829) to handle in a thoroughly
| |
| scientific manner (see [[convergence of Fourier series]]). Dirichlet's treatment (''[[Crelle]]'', 1829), of trigonometric series was the subject of criticism and improvement by
| |
| Riemann (1854), Heine, [[Rudolf Lipschitz|Lipschitz]], [[Ludwig Schläfli|Schläfli]], and
| |
| [[Paul du Bois-Reymond|du Bois-Reymond]]. Among other prominent contributors to the theory of
| |
| trigonometric and Fourier series were [[Ulisse Dini|Dini]], [[Charles Hermite|Hermite]], [[Georges Henri Halphen|Halphen]],
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| Krause, Byerly and [[Paul Émile Appell|Appell]].
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| | |
| ==Generalizations==
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| ===Asymptotic series===
| |
| [[Asymptotic series]], otherwise [[asymptotic expansion]]s, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical [[asymptotic series]] reaches its best approximation; if more terms are included, most such series will produce worse answers.
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| | |
| ===Divergent series===
| |
| {{Main|Divergent series}}
| |
| Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A [[summability method]] is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include [[Cesàro summation]], (''C'',''k'') summation, [[Abel summation]], and [[Borel summation]], in increasing order of generality (and hence applicable to increasingly divergent series).
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| | |
| A variety of general results concerning possible summability methods are known. The [[Silverman–Toeplitz theorem]] characterizes ''matrix summability methods'', which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns [[Banach limit]]s.
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| | |
| ===Series in Banach spaces===
| |
| The notion of series can be easily extended to the case of a [[Banach space]]. If ''x''<sub>''n''</sub> is a sequence of elements of a Banach space ''X'', then the series Σ''x''<sub>''n''</sub> converges to ''x'' ∈ ''X'' if the sequence of partial sums of the series tends to ''x''; to wit,
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| :<math>\biggl\|x - \sum_{n=0}^N x_n\biggr\|\to 0</math>
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| as ''N'' → ∞.
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| | |
| More generally, convergence of series can be defined in any [[abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. Specifically, in this case, Σ''x''<sub>''n''</sub> converges to ''x'' if the sequence of partial sums converges to ''x''.
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| | |
| ===Summations over arbitrary index sets===
| |
| Definitions may be given for sums over an arbitrary index set ''I''. There are two main differences with the usual notion of series: first, there is no specific order given on the set ''I''; second, this set ''I'' may be uncountable.
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| | |
| ====Families of non-negative numbers====
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| When summing a family {''a''<sub>''i''</sub>}, ''i'' ∈ ''I'', of non-negative numbers, one may define
| |
| | |
| :<math>\sum_{i\in I}a_i = \sup \Bigl\{ \sum_{i\in A}a_i\,\big| A \text{ finite, } A \subset I\Bigr\} \in [0, +\infty].</math>
| |
| | |
| When the sum is finite, the set of ''i'' ∈ ''I'' such that ''a<sub>i</sub>'' > 0 is countable. Indeed for every ''n'' ≥ 1, the set <math>\scriptstyle A_n = \{ i \in I \,:\, a_i > 1/n \}</math> is finite, because
| |
| | |
| :<math> \frac 1 n \, \textrm{card}(A_n) \le \sum_{i\in A_n} a_i \le \sum_{i\in I}a_i < \infty.</math>
| |
| | |
| If ''I''  is countably infinite and enumerated as ''I'' = {''i''<sub>0</sub>, ''i''<sub>1</sub>,...} then the above defined sum satisfies
| |
| | |
| :<math>\sum_{i \in I} a_i = \sum_{k=0}^{+\infty} a_{i_k},</math>
| |
| | |
| provided the value ∞ is allowed for the sum of the series.
| |
| | |
| Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the [[counting measure]], which accounts for the many similarities between the two constructions.
| |
| | |
| ====Abelian topological groups====
| |
| Let ''a'' : ''I'' → ''X'', where ''I''  is any set and ''X''  is an [[abelian group|abelian]] [[Hausdorff space|Hausdorff]] [[topological group]]. Let ''F''  be the collection of all [[finite set|finite]] [[subset]]s of ''I''. Note that ''F''  is a [[directed set]] [[Partially ordered set|ordered]] under [[inclusion (mathematics)|inclusion]] with [[union (set theory)|union]] as [[join (mathematics)|join]]. Define the sum ''S''  of the family ''a'' as the limit
| |
| | |
| :<math> S = \sum_{i\in I}a_i = \lim \Bigl\{\sum_{i\in A}a_i\,\big| A\in F\Bigr\}</math>
| |
| | |
| if it exists and say that the family ''a'' is unconditionally summable. Saying that the sum ''S''  is the limit of finite partial sums means that for every neighborhood ''V''  of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I''  such that
| |
| | |
| :<math>S - \sum_{i \in A} a_i \in V, \quad A \supset A_0.</math>
| |
| | |
| Because ''F''  is not [[total order|totally ordered]], this is not a [[limit of a sequence]] of partial sums, but rather of a [[net (mathematics)|net]].<ref name="Bourbaki">{{cite book|title=General Topology: Chapters 1-4 |first=Nicolas |last=Bourbaki |authorlink=Nicolas Bourbaki |year=1998 |publisher=Springer |isbn=9783540642411 |pages=261–270}}</ref><ref name="Choquet">{{cite book|title=Topology |first=Gustave |last=Choquet |authorlink=Gustave Choquet |year=1966 |publisher=Academic Press |isbn=9780121734503 |pages=216–231}}</ref>
| |
| | |
| For every ''W'', neighborhood of 0 in ''X'', there is a smaller neighborhood ''V''  such that ''V'' − ''V'' ⊂ ''W''. It follows that the finite partial sums of an unconditionally summable family ''a<sub>i</sub>'', ''i'' ∈ ''I'', form a ''Cauchy net'', that is: for every ''W'', neighborhood of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I''  such that
| |
| | |
| :<math>\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W, \quad A_1, A_2 \supset A_0.</math>
| |
| | |
| When ''X''  is [[Complete metric space|complete]], a family ''a'' is unconditionally summable in ''X''  if and only if the finite sums satisfy the latter Cauchy net condition. When ''X''  is complete and ''a<sub>i</sub>'', ''i'' ∈ ''I'', is unconditionally summable in ''X'', then for every subset ''J'' ⊂ ''I'', the corresponding subfamily ''a<sub>j</sub>'', ''j'' ∈ ''J'', is also unconditionally summable in ''X''.
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| | |
| When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group ''X'' = '''R'''.
| |
| | |
| If a family ''a'' in ''X''  is unconditionally summable, then for every ''W'', neighborhood of 0 in ''X'', there is a finite subset ''A''<sub>0</sub> of ''I''  such that ''a''<sub>''i''</sub> ∈ ''W''  for every ''i'' not in ''A''<sub>0</sub>. If ''X''  is [[first-countable space|first-countable]], it follows that the set of ''i'' ∈ ''I''  such that ''a<sub>i</sub>'' ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).
| |
| | |
| ====Unconditionally convergent series====
| |
| Suppose that ''I'' = '''N'''. If a family ''a''<sub>''n''</sub>, ''n'' ∈ '''N''', is unconditionally summable in an abelian Hausdorff topological group ''X'', then the series in the usual sense converges and has the same sum,
| |
| | |
| :<math>\sum_{n=0}^\infty a_n = \sum_{n \in \mathbf{N}} a_n.</math>
| |
| | |
| By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑''a''<sub>''n''</sub> is unconditionally summable, then the series remains convergent after any permutation ''σ'' of the set '''N''' of indices, with the same sum,
| |
| | |
| :<math>\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.</math>
| |
| | |
| Conversely, if every permutation of a series ∑''a''<sub>''n''</sub> converges, then the series is unconditionally convergent. When ''X''  is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if ''X''  is a Banach space, this is equivalent to say that for every sequence of signs ''ε''<sub>''n''</sub> = 1 or −1, the series
| |
| | |
| :<math>\sum_{n=0}^\infty \varepsilon_n a_n</math>
| |
| | |
| converges in ''X''. If ''X''  is a Banach space, then one may define the notion of absolute convergence. A series ∑''a''<sub>''n''</sub> of vectors in ''X''  converges absolutely if
| |
| | |
| :<math> \sum_{n \in \mathbf{N}} \|a_n\| < +\infty.</math>
| |
| | |
| 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]] ''α''<sub>0</sub>. One may define by [[transfinite recursion]]:
| |
| | |
| :<math>\sum_{\beta < \alpha + 1} a_\beta = a_{\alpha} + \sum_{\beta < \alpha} a_\beta\,\!</math>
| |
| | |
| and for a limit ordinal ''α'',
| |
| | |
| :<math>\sum_{\beta < \alpha} a_\beta = \lim_{\gamma\to\alpha} \sum_{\beta < \gamma} a_\beta</math>
| |
| | |
| if this limit exists. If all limits exist up to ''α''<sub>0</sub>, then the series converges.
| |
| | |
| ====Examples====
| |
| <ol>
| |
| <li>
| |
| Given a function ''f'' : ''X''→''Y'', with ''Y'' an abelian topological group, define for every ''a'' ∈ ''X''
| |
| | |
| :<math>f_a(x)=
| |
| \begin{cases}
| |
| 0 & x\neq a, \\
| |
| f(a) & x=a, \\
| |
| \end{cases}
| |
| </math>
| |
| | |
| a function whose [[support (mathematics)|support]] is a [[Singleton (mathematics)|singleton]] {''a''}. Then
| |
| | |
| :<math>f=\sum_{a \in X}f_a</math>
| |
| | |
| in the [[topology of pointwise convergence]] (that is, the sum is taken in the infinite product group ''Y''<sup>''X'' </sup>).
| |
| </li>
| |
| <li>
| |
| In the definition of [[partitions of unity]], one constructs sums of functions over arbitrary index set ''I'',
| |
| | |
| :<math> \sum_{i \in I} \varphi_i(x) = 1.</math>
| |
| | |
| 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'', ''i.e.'', for every ''x'' there is a neighborhood of ''x'' in which all but a finite number of functions vanish. Any regularity property of the ''φ<sub>i</sub>'',  such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
| |
| </li>
| |
| <li>
| |
| On the [[first uncountable ordinal]] ω<sub>1</sub> viewed as a topological space in the [[order topology]], the constant function ''f'': [0,ω<sub>1</sub>) → [0,ω<sub>1</sub>] given by ''f''(α) = 1 satisfies
| |
| | |
| :<math>\sum_{\alpha\in[0,\omega_1)}f(\alpha) = \omega_1</math>
| |
| | |
| (in other words, ω<sub>1</sub> copies of 1 is ω<sub>1</sub>) only if one takes a limit over all ''countable'' partial sums, rather than finite partial sums. This space is not separable.
| |
| </li>
| |
| | |
| </ol>
| |
| | |
| ==See also==
| |
| *[[Convergent series]]
| |
| *[[Convergence tests]]
| |
| *[[Sequence transformation]]
| |
| *[[Infinite product]]
| |
| *[[Infinite expression (mathematics)|Infinite expression]]
| |
| *[[Continued fraction]]
| |
| *[[Iterated binary operation]]
| |
| *[[List of mathematical series]]
| |
| *[[Prefix sum]]
| |
| *[[Taylor series|Series expansion]]
| |
| *[[Infinite compositions of analytic functions]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * [[Thomas John I'Anson Bromwich|Bromwich, T.J.]] ''An Introduction to the Theory of Infinite Series'' MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.
| |
| * {{cite journal
| |
| | doi = 10.1073/pnas.36.3.192
| |
| | last1 = Dvoretzky
| |
| | first1 = Aryeh
| |
| | last2 = Rogers
| |
| | first2 = C. Ambrose
| |
| | title = Absolute and unconditional convergence in normed linear spaces
| |
| | journal = Proc. Nat. Acad. Sci. U. S. A.
| |
| | volume = 36
| |
| | issue = 3
| |
| | year = 1950
| |
| | pages = 192–197
| |
| }} {{MathSciNet|id=0033975}}
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| | |
| ==External links==
| |
| {{Commons category|Series (mathematics)}}
| |
| * {{springer|title=Series|id=p/s084670}}
| |
| * [http://www.math.odu.edu/~bogacki/citat/series/index.html Infinite Series Tutorial]
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| | |
| {{Series (mathematics)}}
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| {{DEFAULTSORT:Series (Mathematics)}}
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| [[Category:Calculus]]
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| [[Category:Mathematical series| ]]
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