Series (mathematics)

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. 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 in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

In modern terminology, any ordered infinite sequence \((a_1,a_2,a_3,\ldots)\) of terms, whether those terms are numbers, functions, matrices, or anything else that can be added, defines a series, which is the addition of the ⁠\(a_i\)⁠ one after the other. To emphasize that there are an infinite number of terms, series are often also called infinite series to contrast with finite series, a term sometimes used for finite sums. Series are represented by an expression like \[a_1+a_2+a_3+\cdots,\] or, using capital-sigma summation notation, \[\sum_{i=1}^\infty a_i.\]

The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series. This value is the limit as ⁠\(n\)⁠ tends to infinity of the finite sums of the ⁠\(n\)⁠ first terms of the series if the limit exists. These finite sums are called the partial sums of the series. Using summation notation, \[\sum_{i=1}^\infty a_i = \lim_{n\to\infty}\, \sum_{i=1}^n a_i,\] if it exists. When the limit exists, the series is convergent or summable and also the sequence \((a_1,a_2,a_3,\ldots)\) is summable, and otherwise, when the limit does not exist, the series is divergent.

The expression \(\sum_{i=1}^\infty a_i\) denotes both the series, the implicit process of adding the terms one after the other indefinitely, and, if the series is convergent, the sum of the series, the explicit limit of the process. This is a generalization of the similar convention of denoting by \(a+b\) both the addition, the process of adding, and its result, the sum of ⁠\(a\)⁠ and ⁠\(b\)⁠.

Condensed: the full section is in Wikipedia.

A series or, redundantly, an infinite series, is an infinite sum. It is often represented as \[a_0 + a_1 + a_2 + \cdots \quad \text{or} \quad a_1 + a_2 + a_3 + \cdots,\] where the terms \(a_k\) are the members of a sequence of numbers, functions, or anything else that can be added. A series may also be represented with capital-sigma notation: \[\sum_{k=0}^{\infty} a_k \qquad \text{or} \qquad \sum_{k=1}^{\infty} a_k .\]

It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the ⁠\(n\)⁠th term as a function of ⁠\(n\)⁠: \[a_0 + a_1 + a_2 + \cdots + a_n +\cdots \quad \text{ or } \quad f(0) + f(1) + f(2) + \cdots + f(n) + \cdots.\] For example, Euler's number can be defined with the series \[\sum_{n=0}^\infty \frac 1{n!}=1+1+\frac12 +\frac 16 +\cdots + \frac 1{n!}+\cdots,\] where \(n!\) denotes the product of the \(n\) first positive integers, and \(0!\) is conventionally equal to \(1.\)

Given a series \(s=\sum_{k=0}^\infty a_k\), its ⁠\(n\)⁠th partial sum is \[s_n = \sum_{k=0}^{n} a_k = a_0 + a_1 + \cdots + a_n .\]

Some authors directly identify a series with its sequence of partial sums. Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements, \[a_n = s_{n} - s_{n-1}.\]

Partial summation of a sequence is an example of a linear sequence transformation, and it is also known as the prefix sum in computer science. The inverse transformation for recovering a sequence from its partial sums is the finite difference, another linear sequence transformation.

Partial sums of series sometimes have simpler closed form expressions, for instance an arithmetic series has partial sums \[s_n = \sum_{k=0}^{n} \left(a + kd\right) = a + (a + d) + (a + 2d) + \cdots + (a + nd) = (n+1)\bigl(a + \tfrac12 n d\bigr),\] and a geometric series has partial sums \[s_n = \sum_{k=0}^{n} ar^k = a + ar + ar^2 + \cdots + ar^n = a\frac{1 - r^{n+1}}{1 - r}\] if ⁠\(r \neq 1\)⁠ or simply ⁠\(s_n = a(n+1)\)⁠ if ⁠\(r = 1\)⁠.

Strictly speaking, a series is said to converge, to be convergent, or to be summable when the sequence of its partial sums has a limit. When the limit of the sequence of partial sums does not exist, the series diverges or is divergent. When the limit of the partial sums exists, it is called the sum of the series or value of the series: \[\sum_{k = 0}^\infty a_k = \lim_{n\to\infty} \sum_{k=0}^n a_k = \lim_{n\to\infty} s_n.\] A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms. When the sum exists, the difference between the sum of a series and its \(n\)th partial sum, \(s - s_n = \sum_{k=n+1}^\infty a_k,\) is known as the \(n\)th truncation error of the infinite series.

An example of a convergent series is the geometric series \[1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8} + \cdots + \frac{1}{2^k} + \cdots.\]

It can be shown by algebraic computation that each partial sum \(s_n\) is \[\sum_{k=0}^n \frac 1{2^k} = 2-\frac 1{2^n}.\] As one has \[\lim_{n \to \infty} \left(2-\frac 1{2^n}\right) =2,\] the series is convergent and converges to ⁠\(2\)⁠ with truncation errors \(1 / 2^n\).

By contrast, the geometric series \[\sum_{k = 0}^\infty 2^k\] is divergent in the real numbers. However, it is convergent in the extended real number line, with \(+\infty\) as its limit and \(+\infty\) as its truncation error at every step.

When a series's sequence of partial sums is not easily calculated and evaluated for convergence directly, convergence tests can be used to prove that the series converges or diverges.

In ordinary finite summations, terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of the associativity of addition. \(a_0 + a_1 + a_2 = {}\)\(a_0 + (a_1 + a_2) = {}\)\((a_0 + a_1) + a_2.\) Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum of \(a_0 + a_1 + a_2 + \cdots\) may not equal the sum of \(a_0 + (a_1 + a_2) + {}\)\((a_3 + a_4) + \cdots.\)

For example, Grandi's series ⁠\(1-1+1-1+ \cdots\)⁠ has a sequence of partial sums that alternates back and forth between ⁠\(1\)⁠ and ⁠\(0\)⁠ and does not converge. Grouping its elements in pairs creates the series \((1 - 1) + (1 - 1) + (1 - 1) + \cdots = {}\)\(0 + 0 + 0 + \cdots,\) which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series \(1 + (- 1 + 1) + {}\)\((- 1 + 1) + \cdots = {}\)\(1 + 0 + 0 + \cdots,\) which has partial sums equal to one for every term and thus sums to one, a different result.

In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a subsequence of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied in Oresme's proof of the divergence of the harmonic series, and it is the basis for the general Cauchy condensation test.

In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of the commutativity of addition. \(a_0 + a_1 + a_2 = {}\)\(a_0 + a_2 + a_1 = {}\)\(a_2 + a_1 + a_0.\) Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.

However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called conditionally convergent series. Those that converge to the same value regardless of rearrangement are called unconditionally convergent series.

For series of real numbers and complex numbers, a series \(a_0 + a_1 + a_2 + \cdots\) is unconditionally convergent if and only if the series summing the absolute values of its terms, \(|a_0| + |a_1| + |a_2| + \cdots,\) is also convergent, a property called absolute convergence. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem.

A historically important example of conditional convergence is the alternating harmonic series,

Condensed: the full section is in Wikipedia.

The addition of two series \(a_0 + a_1 + a_2 + \cdots\) and \(b_0 + b_1 + b_2 + \cdots\) is given by the termwise sum \((a_0 + b_0) + (a_1 + b_1) + (a_2 + b_2) + \cdots \,\), or, in summation notation, \[\sum_{k=0}^{\infty} a_k + \sum_{k=0}^{\infty} b_k = \sum_{k=0}^{\infty} a_k + b_k.\]

Using the symbols \(s_{a, n}\) and \(s_{b, n}\) for the partial sums of the added series and \(s_{a + b, n}\) for the partial sums of the resulting series, this definition implies the partial sums of the resulting series follow \(s_{a + b, n} = s_{a, n} + s_{b, n}.\) Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfies \[\lim_{n \rightarrow \infty} s_{a + b, n} = \lim_{n \rightarrow \infty} (s_{a, n} + s_{b, n}) = \lim_{n \rightarrow \infty} s_{a, n} + \lim_{n \rightarrow \infty} s_{b , n},\] when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times \(-1\) will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.

For series of real numbers or complex numbers, series addition is associative, commutative, and invertible. Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of an abelian group and also gives the sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group.

The product of a series \(a_0 + a_1 + a_2 + \cdots\) with a constant number \(c\), called a scalar in this context, is given by the termwise product \(ca_0 + ca_1 + ca_2 + \cdots\), or, in summation notation,

\[c\sum_{k=0}^{\infty} a_k = \sum_{k=0}^{\infty} ca_k.\]

Using the symbols \(s_{a, n}\) for the partial sums of the original series and \(s_{ca, n}\) for the partial sums of the series after multiplication by \(c\), this definition implies that \(s_{ca, n} = c s_{a, n}\) for all \(n,\) and therefore also \(\lim_{n \rightarrow \infty} s_{ca, n} = c \lim_{n \rightarrow \infty} s_{a, n},\)when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent.

Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and it distributes over series addition.

In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of a real vector space. Similarly, one gets complex vector spaces for series and convergent series of complex numbers. All these vector spaces are infinite dimensional.

The multiplication of two series \(a_0 + a_1 + a_2 + \cdots\) and \(b_0 + b_1 + b_2 + \cdots\) to generate a third series \(c_0 + c_1 + c_2 + \cdots\), called the Cauchy product, can be written in summation notation \[\biggl( \sum_{k=0}^{\infty} a_k \biggr) \cdot \biggl( \sum_{k=0}^{\infty} b_k \biggr) = \sum_{k=0}^{\infty} c_k = \sum_{k=0}^{\infty} \sum_{j=0}^{k} a_{j} b_{k-j},\] with each \(c_k = \sum_{j=0}^{k} a_{j} b_{k-j} = {}\!\)\(\!a_0 b_k + a_1 b_{k-1} + \cdots + a_{k-1} b_1 + a_k b_0.\) Here, the convergence of the partial sums of the series \(c_0 + c_1 + c_2 + \cdots\) is not as simple to establish as for addition. However, if both series \(a_0 + a_1 + a_2 + \cdots\) and \(b_0 + b_1 + b_2 + \cdots\) are absolutely convergent series, then the series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series, \[\lim_{n \rightarrow \infty} s_{c, n} = \left(\, \lim_{n \rightarrow \infty} s_{a, n} \right) \cdot \left(\, \lim_{n \rightarrow \infty} s_{b , n} \right).\]

Series multiplication of absolutely convergent series of real numbers and complex numbers is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of a commutative ring, and together with scalar multiplication as well, the structure of a commutative algebra; these operations also give the sets of all series of real numbers or complex numbers the structure of an associative algebra.

• 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). For example: \[1 + {1 \over 2} + {1 \over 4} + {1 \over 8} + {1 \over 16} + \cdots=\sum_{n=0}^\infty{1 \over 2^n} = 2.\] In general, a geometric series with initial term \(a\) and common ratio \(r\), \(\sum_{n=0}^\infty a r^n,\) converges if and only if \(|r| < 1\), in which case it converges to \({a \over 1 - r}\).
• The harmonic series is the series \[1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots = \sum_{n=1}^\infty {1 \over n}.\] The harmonic series is divergent.
• An alternating series is a series where terms alternate signs. Examples: \[1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots = \sum_{n=1}^\infty {\left(-1\right)^{n-1} \over n} = \ln(2),\] the alternating harmonic series, and \[-1+\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \cdots = \sum_{n=1}^\infty \frac{\left(-1\right)^n}{2n-1} = -\frac{\pi}{4},\] the Leibniz formula for \(\pi.\)
• A telescoping series \[\sum_{n=1}^\infty \left(b_n-b_{n+1}\right)\] converges if the sequence ⁠\(b_n\)⁠ converges to a limit ⁠\(L\)⁠ as ⁠\(n\)⁠ goes to infinity. The value of the series is then ⁠\(b_1 - L\)⁠.
• An arithmetico-geometric series is a series that has terms which are each the product of an element of an arithmetic progression with the corresponding element of a geometric progression. Example: \[3 + {5 \over 2} + {7 \over 4} + {9 \over 8} + {11 \over 16} + \cdots=\sum_{n=0}^\infty{(3+2n) \over 2^n}.\]
• The Dirichlet series \[\sum_{n=1}^\infty\frac{1}{n^p}\] converges for ⁠\(p>1\)⁠ and diverges for ⁠\(p \leq 1\)⁠, which can be shown with the integral test for convergence described below in convergence tests. As a function of ⁠\(p\)⁠, the sum of this series is Riemann's zeta function.
• Hypergeometric series: \[_pF_q \left[ \begin{matrix}a_1, a_2, \dotsc, a_p \\ b_1, b_2, \dotsc, b_q \end{matrix}; z \right] := \sum_{n=0}^{\infty} \frac{\prod_{r=1}^{p} (a_r)_n}{\prod_{s=1}^{q} (b_s)_n} \frac{z^n}{n!}\] and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and mathematical physics.
• There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series, \[\sum_{n=1}^\infty \frac{1}{n^{3}\sin^{2} n},\] 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, ... (sequence A046947 in the OEIS). These are integers ⁠\(n\)⁠ that are close to \(m\pi\) for some integer ⁠\(m\)⁠, so that \(\sin n\) is close to \(\sin m\pi = 0\) and its reciprocal is large.

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\]

\[4\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots = \pi\]

\[\sum_{n = 0}^\infty \frac{(-1)^n}{n!} = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots = \frac{1}{e}\]

\[\sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e\]

One of the simplest tests for convergence of a series, applicable to all series, is the vanishing condition or ⁠\(n\)⁠th-term test: If \(\lim_{n \to \infty} a_n \neq 0\), then the series diverges; if \(\lim_{n \to \infty} a_n = 0\), then the test is inconclusive.

When every term of a series is a non-negative real number, for instance when the terms are the absolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound for a series or for the absolute values of its terms is an effective way to prove convergence or absolute convergence of a series.

For example, the series \(1 + \frac14 + \frac19 + \cdots + \frac1{n^2} + \cdots\,\)is convergent and absolutely convergent because \(\frac1{n^2} \le \frac1{n-1} - \frac1n\) for all \(n \geq 2\) and a telescoping sum argument implies that the partial sums of the series of those non-negative bounding terms are themselves bounded above by 2. The exact value of this series is \(\frac16\pi^2\); see Basel problem.

This type of bounding strategy is the basis for general series comparison tests. First is the general direct comparison test: For any series \(\sum a_n\), If \(\sum b_n\) is an absolutely convergent series such that \(\left\vert a_n \right\vert \leq C \left\vert b_n \right\vert\) for some positive real number \(C\) and for sufficiently large \(n\), then \(\sum a_n\) converges absolutely as well. If \(\sum \left\vert b_n \right\vert\) diverges, and \(\left\vert a_n \right\vert \geq \left\vert b_n \right\vert\) for all sufficiently large \(n\), then \(\sum a_n\) also fails to converge absolutely, although it could still be conditionally convergent, for example, if the \(a_n\) alternate in sign. Second is the general limit comparison test: If \(\sum b_n\) is an absolutely convergent series such that \(\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert \leq \left\vert \tfrac{b_{n+1}}{b_{n}} \right\vert\) for sufficiently large \(n\), then \(\sum a_n\) converges absolutely as well. If \(\sum \left| b_n \right|\) diverges, and \(\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert \geq \left\vert \tfrac{b_{n+1}}{b_{n}} \right\vert\) for all sufficiently large \(n\), then \(\sum a_n\) also fails to converge absolutely, though it could still be conditionally convergent if the \(a_n\) vary in sign.

Using comparisons to geometric series specifically, those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms. First is the ratio test: if there exists a constant \(C < 1\) such that \(\left\vert \tfrac{a_{n+1}}{a_{n}} \right\vert < C\) for all sufficiently large \(n\), then \(\sum a_{n}\) 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. Second is the root test: if there exists a constant \(C < 1\) such that \(\textstyle \left\vert a_{n} \right\vert^{1/n} \leq C\) for all sufficiently large \(n\), then \(\sum a_{n}\) converges absolutely.

Alternatively, using comparisons to series representations of integrals specifically, one derives the integral test: if \(f(x)\) is a positive monotone decreasing function defined on the interval \([1,\infty)\) then for a series with terms \(a_n = f(n)\) for all \(n\), \(\sum a_{n}\) converges if and only if the integral \(\int_{1}^{\infty} f(x) \, dx\) is finite. Using comparisons to flattened-out versions of a series leads to Cauchy's condensation test: if the sequence of terms \(a_{n}\) is non-negative and non-increasing, then the two series \(\sum a_{n}\) and \(\sum 2^{k} a_{(2^{k})}\) are either both convergent or both divergent.