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Sequence
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"Sequential" redirects here. For the manual transmission, see Sequential manual transmission. For other uses, see Sequence (disambiguation).
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n). The position of an element in a sequence is its rank or index; it is the integer from which the element is the image. It depends on the context or of a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the nth element of the sequence is denoted by this symbol with n as subscript; for example, the nth element of the Fibonacci sequence is generally denoted Fn.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded.
Contents
[hide]
1 Examples and notation
1.1 Examples
1.2 Indexing
1.3 Defining a sequence by recursion
2 Formal definition and basic properties
2.1 Formal definition
2.2 Finite and infinite
2.3 Increasing and decreasing
2.4 Bounded
2.5 Subsequences
2.6 Other types of sequences
3 Limits and convergence
3.1 Formal definition of convergence
3.2 Applications and important results
3.3 Cauchy sequences
3.4 Infinite limits
4 Series
5 Use in other fields of mathematics
5.1 Topology
5.1.1 Product topology
5.2 Analysis
5.2.1 Sequence spaces
5.3 Linear algebra
5.4 Abstract algebra
5.4.1 Free monoid
5.4.2 Exact sequences
5.4.3 Spectral sequences
5.5 Set theory
5.6 Computing
5.7 Streams
6 See also
7 References
8 External links
Examples and notation[edit]
A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.
There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive odd integers can be written (1, 3, 5, 7, ...). Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.
Examples[edit]
A tiling with squares whose sides are successive Fibonacci numbers in length.
The prime numbers are the natural numbers bigger than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics and specifically in number theory.
The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).
For a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences.
Other examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. The sequence (.9, .99, .999, .9999, ...) approaches the number 1. In fact, every real number can be written as the limit of a sequence of rational numbers, e.g. via its decimal expansion. For instance, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). A related sequence is the sequence of decimal digits of π, i.e. (3, 1, 4, 1, 5, 9, ...). This sequence does not have any pattern that is easily discernible by eye, unlike the preceding sequence, which is increasing.
Indexing[edit]
Other notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not have a pattern such as the digits of π. One such notation is to write down a general formula for computing the nth term as a function of n, enclose it in parentheses, and include a subscript indicating the range of values that n can take. For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\displaystyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\displaystyle (n^{2})_{n\in \mathbb {N} }} . The variable n is called an index, and the set of values that it can take is called the index set.
It is often useful to combine this notation with the technique of treating the elements of a sequence as variables. This yields expressions like ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} , which denotes a sequence whose nth element is given by the variable a n {\displaystyle a_{n}} . For example:
a 1 = 1 st element of ( a n ) n ∈ N a 2 = 2 nd element a 3 = 3 rd element ⋮ a n − 1 = ( n − 1 ) th element a n = n th element a n + 1 = ( n + 1 ) th element ⋮ {\displaystyle {\begin{aligned}a_{1}&=1{\text{st element of }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{nd element }}\\a_{3}&=3{\text{rd element }}\\&\vdots \\a_{n-1}&=(n-1){\text{th element}}\\a_{n}&=n{\text{th element}}\\a_{n+1}&=(n+1){\text{th element}}\\&\vdots \end{aligned}}}
Note that we can consider multiple sequences at the same time by using different variables; e.g. ( b n ) n ∈ N {\displaystyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} . We can even consider a sequence of sequences: ( ( a m , n ) n ∈ N ) m ∈ N {\displaystyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose mth term is the sequence ( a m , n ) n ∈ N {\displaystyle (a_{m,n})_{n\in \mathbb {N} }} .
An alternative to writing the domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ( k 2 ) k = 1 10 {\displaystyle (k^{2})_{k=1}^{10}} denotes the ten-term sequence of squares ( 1 , 4 , 9 , . . . , 100 ) {\displaystyle (1,4,9,...,100)} . The limits ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } are allowed, but they do not represent valid values for the index, only the supremum or infimum of such values, respectively. For example, the sequence ( a n ) n = 1 ∞ {\displaystyle (a_{n})_{n=1}^{\infty }} is the same as the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} , and does not contain an additional term "at infinity". The sequence ( a n ) n = − ∞ ∞ {\displaystyle (a_{n})_{n=-\infty }^{\infty }} is a bi-infinite sequence, and can also be written as ( . . . , a − 1 , a 0 , a 1 , a 2 , . . . ) {\displaystyle (...,a_{-1},a_{0},a_{1},a_{2},...)} .
In cases where the set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ( a k ) {\displaystyle (a_{k})} for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in
( a k ) k = 0 ∞ = ( a 0 , a 1 , a 2 , . . . ) . {\displaystyle (a_{k})_{k=0}^{\infty }=(a_{0},a_{1},a_{2},...).}
In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways.
( 1 , 9 , 25 , . . . ) {\displaystyle (1,9,25,...)}
( a 1 , a 3 , a 5 , . . . ) , a k = k 2 {\displaystyle (a_{1},a_{3},a_{5},...),\qquad a_{k}=k^{2}}
( a 2 k − 1 ) k = 1 ∞ , a k = k 2 {\displaystyle (a_{2k-1})_{k=1}^{\infty },\qquad a_{k}=k^{2}}
( a k ) k = 1 ∞ , a k = ( 2 k − 1 ) 2 {\displaystyle (a_{k})_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}
( ( 2 k − 1 ) 2 ) k = 1 ∞ {\displaystyle ((2k-1)^{2})_{k=1}^{\infty }}
Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be the natural numbers. Note that in the second and third bullets, there is a well-defined sequence ( a k ) k = 1 ∞ {\displaystyle (a_{k})_{k=1}^{\infty }} , but it is not the same as the sequence denoted by the expression.
Defining a sequence by recursion[edit]
Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. This is in contrast to the definition of sequence elements as a function of their position.
To define a sequence by recursion, one needs a rule to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by the rule. The principle of mathematical induction can be used to prove that in this case, there is exactly one sequence that satisfies both the recursion rule and the initial conditions. Induction can also be used to prove properties about a sequence, especially for sequences whose most natural description is recursive.
The Fibonacci sequence can be defined using a recursive rule along with two initial elements. The rule is that each element is the sum of the previous two elements, and the first two elements are 0 and 1.
a n = a n − 1 + a n − 2 {\displaystyle a_{n}=a_{n-1}+a_{n-2}} , with a 0 = 0 {\displaystyle a_{0}=0} and a 1 = 1 {\displaystyle a_{1}=1} .
The first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A more complicated example of a sequence that is defined recursively is Recaman's sequence. We can define Recaman's sequence by
a 0 = 0 {\displaystyle a_{0}=0} and a n = a n − 1 − n {\displaystyle a_{n}=a_{n-1}-n} , if the result is positive and not already in the list. Otherwise, a n = a n − 1 + n {\displaystyle a_{n}=a_{n-1}+n} .
Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quite complicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order, i.e. (2, 3, 5, 7, 11, 13, 17, ...).
Many sequences have the property that each element of a sequence can be computed from the previous element. In this case, there is some function f such that for all n, a n + 1 = f ( a n ) {\displaystyle a_{n+1}=f(a_{n})} .
Formal definition and basic properties[edit]
There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the definitions and notations introduced below.
Formal definition[edit]
For the purposes of this article, we define a sequence to be a function whose domain is a convex subset of the set of integers. This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. The narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Many authors also impose a requirement on the codomain of a function before calling it a sequence, by requiring it to be the set R of real numbers,[1] the set C of complex numbers,[2] or a topological space.[3]
Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, i.e. an rather than f(n). There are terminological differences as well: the value of a sequence at the input 1 is called the "first element" of the sequence, the value at 2 is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f, a sequence abstracted from its input is usually written by a notation such as ( a n ) n ∈ A {\displaystyle (a_{n})_{n\in A}} , or just as ( a n ) {\displaystyle (a_{n})} . Here A is the domain, or index set, of the sequence.
Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets. A net is a function from a (possibly uncountable) directed set to a topological space. The notational conventions for sequences normally apply to nets as well.
Finite and infinite[edit]
See also: ω-language
The length of a sequence is defined as the number of terms in the sequence.
A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements.
Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( …, −4, −2, 0, 2, 4, 6, 8… ), is bi-infinite. This sequence could be denoted ( 2 n ) n = − ∞ ∞ {\displaystyle (2n)_{n=-\infty }^{\infty }} .
Increasing and decreasing[edit]
A sequence is said to be monotonically increasing, if each term is greater than or equal to the one before it. For example, the sequence ( a n ) n = 1 ∞ {\displaystyle (a_{n})_{n=1}^{\infty }} is monotonically increasing if and only if an+1 ≥ {\displaystyle \geq } an for all n ∈ N. If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically decreasing, if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasing, if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function.
The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.
Bounded[edit]
If the sequence of real numbers (an) is such that all the terms are less than some real number M, then the sequence is said to be bounded from above. In less words, this means that there exists M such that for all n, an ≤ M. Any such M is called an upper bound. Likewise, if, for some real m, an ≥ m for all n greater than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded.
Subsequences[edit]
A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of the positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved.
Formally, a subsequence of the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is any sequence of the form ( a n k ) k ∈ N {\displaystyle (a_{n_{k}})_{k\in \mathbb {N} }} , where ( n k ) k ∈ N {\displaystyle (n_{k})_{k\in \mathbb {N} }} is a strictly increasing sequence of positive integers.
Other types of sequences[edit]
Some other types of sequences that are easy to define include:
An integer sequence is a sequence whose terms are integers.
A polynomial sequence is a sequence whose terms are polynomials.
A positive integer sequence is sometimes called multiplicative, if anm = an am for all pairs n, m such that n and m are coprime.[4] In other instances, sequences are often called multiplicative, if an = na1 for all n. Moreover, a multiplicative Fibonacci sequence[5] satisfies the recursion relation an = an−1 an−2.
Limits and convergence[edit]
Main article: Limit of a sequence
The plot of a convergent sequence (an) is shown in blue. From the graph we can see that the sequence is converging to the limit zero as n increases.
An important property of a sequence is convergence. A sequence converges, it converges to a particular value known as the limit. If a sequence converges to some limit, then it is convergent. A sequence that does not converge is divergent.
Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value L {\displaystyle L} (called the limit of the sequence), and they become and remain arbitrarily close to L {\displaystyle L} , meaning that given a real number d {\displaystyle d} greater than zero, all but a finite number of the elements of the sequence have a distance from L {\displaystyle L} less than d {\displaystyle d} .
For example, the sequence a n = n + 1 2 n 2 {\displaystyle a_{n}={\frac {n+1}{2n^{2}}}} shown to the right converges to the value 0. On the other hand, the sequences b n = n 3 {\displaystyle b_{n}=n^{3}} (which begins 1, 8, 27, …) and c n = ( − 1 ) n {\displaystyle c_{n}=(-1)^{n}} (which begins -1, 1, -1, 1, …) are both divergent.
If a sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ( a n ) {\displaystyle (a_{n})} is normally denoted lim n → ∞ a n {\displaystyle \lim _{n\to \infty }a_{n}} . If ( a n ) {\displaystyle (a_{n})} is a divergent sequence, then the expression lim n → ∞ a n {\displaystyle \lim _{n\to \infty }a_{n}} is meaningless.
Formal definition of convergence[edit]
A sequence of real numbers ( a n ) {\displaystyle (a_{n})} converges to a real number L {\displaystyle L} if, for all ε > 0 {\displaystyle \varepsilon >0} , there exists a natural number N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} we have[1]
| a n − L | < ε . {\displaystyle |a_{n}-L|<\varepsilon .}
If ( a n ) {\displaystyle (a_{n})} is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that | ⋅ | {\displaystyle |\cdot |} denotes the complex modulus, i.e. | z | = z ∗ z {\displaystyle |z|={\sqrt {z^{*}z}}} . If ( a n ) {\displaystyle (a_{n})} is a sequence of points in a metric space, then the formula can be used to define convergence, if the expression | a n − L | {\displaystyle |a_{n}-L|} is replaced by the expression dist ( a n , L ) {\displaystyle {\text{dist}}(a_{n},L)} , which denotes the distance between a n {\displaystyle a_{n}} and L {\displaystyle L} .
Applications and important results[edit]
Let ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} be convergent sequences, and let ( c n ) {\displaystyle (c_{n})} be a sequence that is not necessarily convergent. Then the following limits exist, and can be computed as follows:[1][6]
lim n → ∞ ( a n ± b n ) = lim n → ∞ a n ± lim n → ∞ b n {\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}
lim n → ∞ c a n = c lim n → ∞ a n {\displaystyle \lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}} for all c
lim n → ∞ ( a n b n ) = ( lim n → ∞ a n ) ( lim n → ∞ b n ) {\displaystyle \lim _{n\to \infty }(a_{n}b_{n})=(\lim _{n\to \infty }a_{n})(\lim _{n\to \infty }b_{n})}
lim n → ∞ a n b n = lim n → ∞ a n lim n → ∞ b n {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\frac {\lim \limits _{n\to \infty }a_{n}}{\lim \limits _{n\to \infty }b_{n}}}} provided lim n → ∞ b n ≠ 0 {\displaystyle \lim _{n\to \infty }b_{n}\neq 0}
lim n → ∞ a n p = [ lim n → ∞ a n ] p {\displaystyle \lim _{n\to \infty }a_{n}^{p}=\left[\lim _{n\to \infty }a_{n}\right]^{p}} for all p > 0 {\displaystyle p>0}
Moreover:
If an ≤ bn for all n greater than some N, then lim n → ∞ a n ≤ lim n → ∞ b n {\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}} . (Note that, if the inequalities are replaced by strict inequalities then this is false: there are sequences such that an < bn for all n, but lim n → ∞ a n = lim n → ∞ b n {\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}} .)
(Squeeze Theorem) If a n ≤ c n ≤ b n {\displaystyle a_{n}\leq c_{n}\leq b_{n}} for all n > N, and lim n → ∞ a n = lim n → ∞ b n = L {\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L} , then the sequence ( c n ) {\displaystyle (c_{n})} is convergent, and lim n → ∞ c n = L {\displaystyle \lim _{n\to \infty }c_{n}=L} .
If a sequence is bounded and monotonic then it is convergent.
A sequence is convergent if and only if all of its subsequences are convergent.
Cauchy sequences[edit]
Main article: Cauchy sequence
The plot of a Cauchy sequence (Xn), shown in blue, as Xn versus n. In the graph the sequence appears to be converging to a limit as the distance between consecutive terms in the sequence gets smaller as n increases. In the real numbers every Cauchy sequence converges to some limit.
A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is Cauchy characterization of convergence for sequences:
A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.
In contrast, there are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by x1 = 1 and xn+1 = xn + 2/xn/2 is Cauchy, but has no rational limit, cf. here. More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers.
Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis.
Infinite limits[edit]
In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If a n {\displaystyle a_{n}} becomes arbitrarily large as n → ∞ {\displaystyle n\to \infty } , we write
lim n → ∞ a n = ∞ . {\displaystyle \lim _{n\to \infty }a_{n}=\infty .}
In this case we say that the sequence diverges, or that it converges to infinity. An example of such a sequence is an = n.
If a n {\displaystyle a_{n}} becomes arbitrarily negative (i.e. negative and large in magnitude) as n → ∞ {\displaystyle n\to \infty } , we write
lim n → ∞ a n = − ∞ {\displaystyle \lim _{n\to \infty }a_{n}=-\infty }
and say that the sequence diverges or converges to negative infinity.
Series[edit]
Main article: Series (mathematics)
A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} or a 1 + a 2 + ⋯ {\displaystyle a_{1}+a_{2}+\cdots } , where ( a n ) {\displaystyle (a_{n})} is a sequence of real or complex numbers. The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the Nth partial sum of the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} is the number
S N = ∑ n = 1 N a n = a 1 + a 2 + ⋯ + a N . {\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}
The partial sums themselves form a sequence ( S N ) N ∈ N {\displaystyle (S_{N})_{N\in \mathbb {N} }} , which is called the sequence of partial sums of the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} . If the sequence of partial sums converges, then we say that the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} is convergent, and the limit lim N → ∞ S N {\displaystyle \lim _{N\to \infty }S_{N}} is called the value of the series. The same notation is used to denote a series and its value, i.e. we write ∑ n = 1 ∞ a n = lim N → ∞ S N {\displaystyle \sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }S_{N}} .
Use in other fields of mathematics[edit]
Topology[edit]
Sequences play an important role in topology, especially in the study of metric spaces. For instance:
A metric space is compact exactly when it is sequentially compact.
A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences.
A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
A topological space is separable exactly when there is a dense sequence of points.
Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems to spaces without metrics.
Product topology[edit]
The topological product of a sequence of topological spaces is the cartesian product of those spaces, equipped with a natural topology called the product topology.
More formally, given a sequence of spaces ( X i ) i ∈ N {\displaystyle (X_{i})_{i\in \mathbb {N} }} , the product space
X := ∏ i ∈ I X i , {\displaystyle X:=\prod _{i\in I}X_{i},}
is defined as the set of all sequences ( x i ) i ∈ N {\displaystyle (x_{i})_{i\in \mathbb {N} }} such that for each i, x i {\displaystyle x_{i}} is an element of X i {\displaystyle X_{i}} . The canonical projections are the maps pi : X → Xi defined by the equation p i ( ( x j ) j ∈ N ) = x i {\displaystyle p_{i}((x_{j})_{j\in \mathbb {N} })=x_{i}} . Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.
Analysis[edit]
In analysis, when talking about sequences, one will generally consider sequences of the form
( x 1 , x 2 , x 3 , … ) or ( x 0 , x 1 , x 2 , … ) {\displaystyle (x_{1},x_{2},x_{3},\dots ){\text{ or }}(x_{0},x_{1},x_{2},\dots )}
which is to say, infinite sequences of elements indexed by natural numbers.
It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.
The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.
Sequence spaces[edit]
Main article: Sequence space
A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K, where K is either the field of real numbers or the field of complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space.
Linear algebra[edit]
Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.
Abstract algebra[edit]
Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.
Free monoid[edit]
Main article: Free monoid
If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the finite sequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroup A+ is the subsemigroup of A* containing all elements except the empty sequence.
Exact sequences[edit]
Main article: Exact sequence
In the context of group theory, a sequence
G 0 → f 1 G 1 → f 2 G 2 → f 3 ⋯ → f n G n {\displaystyle G_{0}\;{\xrightarrow {f_{1}}}\;G_{1}\;{\xrightarrow {f_{2}}}\;G_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;G_{n}}
of groups and group homomorphisms is called exact, if the image (or range) of each homomorphism is equal to the kernel of the next:
i m ( f k ) = k e r ( f k + 1 ) {\displaystyle \mathrm {im} (f_{k})=\mathrm {ker} (f_{k+1})}
Note that the sequence of groups and homomorphisms may be either finite or infinite.
A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms.
Spectral sequences[edit]
Main article: Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.
Set theory[edit]
An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence.
Computing[edit]
Automata or finite state machines can typically be thought of as directed graphs, with edges labeled using some specific alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word). The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.
Streams[edit]
Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0, 1}∞ of all infinite binary sequences is sometimes called the Cantor space.
An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalization method for proofs.[7]
See also[edit]
Enumeration
On-Line Encyclopedia of Integer Sequences
Recurrence relation
Sequence space
Operations
Cauchy product
Examples
Discrete-time signal
Farey sequence
Fibonacci sequence
Look-and-say sequence
Thue–Morse sequence
Types
±1-sequence
Arithmetic progression
Automatic sequence
Cauchy sequence
Constant-recursive sequence
Geometric progression
Holonomic sequence
Regular sequence
Related concepts
List (computing)
Net (topology) (a generalization of sequences)
Ordinal-indexed sequence
Recursion (computer science)
Set (mathematics)
Tuple
References[edit]
^ Jump up to: a b c Gaughan, Edward. "1.1 Sequences and Convergence". Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.
Jump up ^ Edward B. Saff & Arthur David Snider (2003). "Chapter 2.1". Fundamentals of Complex Analysis. ISBN 01-390-7874-6.
Jump up ^ James R. Munkres. "Chapters 1&2". Topology. ISBN 01-318-1629-2.
Jump up ^ Lando, Sergei K. "7.4 Multiplicative sequences". Lectures on generating functions. AMS. ISBN 0-8218-3481-9.
Jump up ^ Falcon, Sergio. "Fibonacci's multiplicative sequence". International Journal of Mathematical Education in Science and Technology. 34 (2): 310–315. doi:10.1080/0020739031000158362.
Jump up ^ Dawikins, Paul. "Series and Sequences". Paul's Online Math Notes/Calc II (notes). Retrieved 18 December 2012.
Jump up ^ Oflazer, Kemal. "FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY" (PDF). cmu.edu. Carnegie-Mellon University. Retrieved 24 April 2015.
External links[edit]
Look up enumerate or collection in Wiktionary, the free dictionary.
The dictionary definition of sequence at Wiktionary
Hazewinkel, Michiel, ed. (2001) [1994], "Sequence", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
The On-Line Encyclopedia of Integer Sequences
Journal of Integer Sequences (free)
"Sequence". PlanetMath.
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The Life You Save May Be Your Own
From Wikipedia, the free encyclopedia
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"The Life You Save May Be Your Own"
Author
Flannery O'Connor
Country
United States
Language
English
Genre(s)
Southern Gothic
Published in
A Good Man Is Hard to Find
Media type
Print
Publication date
May 12, 1955
"The Life You Save May Be Your Own" is a short story by the American author Flannery O'Connor. It is one of the 10 stories in her short story collection A Good Man Is Hard to Find, published in 1955.
Contents
[hide]
1 Plot summary
2 Adaptation
3 See also
4 References
Plot summary[edit]
An elderly woman and her daughter sit quietly on their porch at sunset when Shiftlet comes walking up the road to their farm. Through carefully selected details, O'Connor reveals that the girl is deaf and mute, that the old woman views Shiftlet as 'a tramp,' and that Shiftlet himself wears a "left coat sleeve that was folded up to show there was only half an arm in it." The two adults exchange curt pleasantries, then Mrs. Crater offers him shelter in exchange for work but warns, "I can't pay." Shiftlet says he has no interest in money, adding that he believes that most people are too concerned with money. Sensing not only a handyman but a suitor for her daughter, Mrs. Crater asks if Shiftlet is married, to which he responds, "Lady, where would you find you an innocent woman today?" Mrs. Crater then makes known her love for her daughter, Lucynell, adding, "She can sweep the floors, cook, wash, feed the chickens, and hoe." Mrs. Crater is clearly offering her daughter's hand to Shiftlet. For the moment, however, he simply decides to stay on the farm and to sleep in the broken-down car. Once Shiftlet moves into the Crater's farm, he fixes a broken fence and hog pen, teaches Lucynell how to speak her first word ("bird"—a recurring symbol in O'Connor's fiction), and, most importantly, repairs the automobile. At this time Mrs. Crater gives her daughter's hand in marriage over to Mr. Shiftlet, but he declines saying, "I can't get married right now, everything you want to do takes money and I ain't got any."
Mrs. Crater, in her desperation to marry off her daughter, offers him a sum of money to marry Lucynell. He then accepts and agrees to marry her. Soon after, the three take the car into town and Lucynell and Shiftlet are married. After the wedding Shiftlet and Lucynell go on their honeymoon. They stop in a restaurant and have dinner. There Lucynell falls asleep. Once she is sound asleep on the counter of the diner, Shiftlet gets up out of his seat and begins to leave. The boy behind the counter looks at the girl and then back at Shiftlet in a confused manner. Seeing how beautiful Lucynell is, the boy exclaims, "She looks like an angel of Gawd". Shiftlet then replies "Hitchhiker" and leaves her at the restaurant. Afterwards Shiftlet "was more depressed than ever" and he "kept his eye out for a hitchhiker." As a storm is breaking in the sky, Shiftlet sees a road sign that reads, "Drive carefully. The life you save may be your own." Shiftlet then offers a ride to a boy who did not even have his thumb out.
Shiftlet tries to make conversation, telling stories about his sweet mother, who is—as the boy at the diner called Lucynell—"an angel of Gawd." But the boy does not buy Shiftlet's sentimentality. "My old woman is a flea bag and yours is a stinking polecat," he snaps, before leaping from the car. Shocked, Shiftlet "felt the rottenness of the world was about to engulf him," exclaiming, "Oh Lord! Break forth and wash the slime from the earth!" The rain finally breaks, with a "guffawing peal of thunder from behind and fantastic raindrops, like tin-can tops, crashed over the rear of Mr. Shiftlet's car." Shiftlet speeds off to Mobile, Alabama.
As in several other O'Connor stories, such as "A Good Man Is Hard to Find" and "Good Country People," in "The Life You Save May Be Your Own" a malevolent stranger intrudes upon the lives of a family with destructive consequences. Tom Shiftlet has been compared to The Misfit in "A Good Man is Hard to Find"; however, Shiftlet remains primarily a comic character and does not embody The Misfit's spiritual dimensions.[1]
Adaptation[edit]
In 1957, the story was adapted into a television production on the Schlitz Playhouse of Stars, starring Gene Kelly.[2]
See also[edit]
"The Love You Save"
References[edit]
Jump up ^ Orvell, Miles. Invisible Parade: The Fiction of Flannery O'Connor. Philadelphia: Temple UP, 1972, p. 135.
Jump up ^ "The Life You Save May Be Your Own". IMDb. Schlitz Playhouse of Stars. 1957.
hide
v
t
e
Flannery O'Connor (works)
Novels
Wise Blood (1952)
The Violent Bear It Away (1960)
Short story
collections
A Good Man Is Hard to Find
(1955)
"A Good Man is Hard to Find"
"The River"
"The Life You Save May Be Your Own"
"A Stroke of Good Fortune"
"A Temple of the Holy Ghost"
"The Artificial Nigger"
"A Circle in the Fire"
"A Late Encounter with the Enemy"
"Good Country People"
"The Displaced Person"
Everything That Rises Must Converge
(1965)
"Everything That Rises Must Converge"
"Greenleaf"
"A View of the Woods"
"The Enduring Chill"
"The Comforts of Home"
"The Lame Shall Enter First"
"Revelation"
"Parker's Back"
"Judgement Day"
Previously unavailable stories
included in The Complete Stories
(1971)
"The Geranium"
"The Barber"
"Wildcat"
"The Crop"
"The Turkey"
"The Train"
"The Peeler"
"The Heart of the Park"
"Enoch and the Gorilla"
"You Can't Be Any Poorer Than Dead"
"The Partridge Festival"
"Why Do the Heathen Rage?"
Related
Flannery O'Connor Childhood Home
Andalusia (estate)
Flannery O'Connor Award for Short Fiction
Retrieved from "https://en.wikipedia.org/w/index.php?title=The_Life_You_Save_May_Be_Your_Own&oldid=848080433"
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List of economics journals
From Wikipedia, the free encyclopedia
Jump to: navigation, search
The following is a list of scholarly journals in economics containing most of the prominent academic journals in economics.
Popular magazines or other publications related to economics, finance, or business are not listed.
Contents:
Top
0–9
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
A[edit]
Affilia
African Journal of Economic Policy
American Economic Journal
American Economic Review
American Journal of Agricultural Economics
Annual Review of Economics
Applied Econometrics and International Development
Applied Economic Perspectives and Policy
Asian Economic Papers
B[edit]
Brookings Papers on Economic Activity
Business Economics
C[edit]
Cambridge Journal of Economics
Canadian Journal of Economics
Carnegie-Rochester Conference Series on Public Policy
Computational Economics
Contemporary Economic Policy
D[edit]
D+C Development and Cooperation
E[edit]
Eastern Economic Journal
Econ Journal Watch
Econometric Theory
Econometrica
Econometrics Journal
Economic and Industrial Democracy
Economic Development and Cultural Change
Economic Geography
Economic History Review
Economic Inquiry
The Economic Journal
Economic Theory
Economica
Economics and Human Biology
Economics Bulletin
Economics Letters
Economics of Governance
De Economist
Education Finance and Policy
Empirical Economic Review
Energy Economics
European Economic Review
European Journal of Health Economics
European Political Economy Review
Explorations in Economic History
F[edit]
Feminist Economics
Fiscal Studies
G[edit]
Games and Economic Behavior
German Economic Review
H[edit]
Health Economics
I[edit]
Industrial and Labor Relations Review
Intereconomics
International Economic Review
International Journal of Central Banking
Investment Analysts Journal
J[edit]
Japanese Economic Review
Journal of Applied Econometrics
Journal of Banking and Finance
Journal of Behavioral Finance
Journal of Business and Economic Statistics
Journal of Common Market Studies
Journal of Comparative Economics
Journal of Competition Law and Economics
Journal of Development Economics
Journal of Econometrics
Journal of Economic Behavior and Organization
Journal of Economic Dynamics and Control
Journal of Economic Education
Journal of Economic History
Journal of Economic Issues
Journal of Economic Literature
Journal of Economic Perspectives
Journal of Economic Theory
Journal of Economics & Management Strategy
Journal of Environmental Economics and Management
Journal of the European Economic Association
Journal of Finance
Journal of Financial and Quantitative Analysis
Journal of Financial Economics
Journal of Health Economics
Journal of Human Capital
Journal of International Economics
Journal of Labor Economics
Journal of Law and Economics
Journal of Macroeconomics
Journal of Mathematical Economics
Journal of Monetary Economics
Journal of Money, Credit and Banking
Journal of Political Economy
Journal of Quantitative Methods
Journal of Regional Science
Journal of Sports Economics
Journal of Wine Economics
L[edit]
Land Economics
M[edit]
Macroeconomic Dynamics
Manchester School
Marine Resource Economics
Mathematical Social Sciences
N[edit]
National Institute Economic Review
New Political Economy
Novi Ekonomist
O[edit]
ORDO – Jahrbuch für die Ordnung von Wirtschaft und Gesellschaft (The Ordo Yearbook of Economic and Social Order)
Oxford Review of Economic Policy
P[edit]
Post-Communist Economies
Public Choice
Q[edit]
Quarterly Journal of Austrian Economics
Quarterly Journal of Economics
R[edit]
Rand Journal of Economics
Real-world economics review
Rethinking Marxism
Review of Agricultural Economics
Review of Austrian Economics
Review of Economic Design
Review of Economic Dynamics
Review of Economic Studies
Review of Economics and Statistics
Review of Environmental Economics and Policy
Review of International Political Economy
Review of Radical Political Economics
S[edit]
Scandinavian Journal of Economics
Science and Society
Scottish Journal of Political Economy
Small Business Economics
Socio-Economic Review
South African Journal of Economic History
South African Journal of Economics
Southern Economic Journal
South Asian Journal of Macroeconomics and Public Finance
W[edit]
Wirtschaftsdienst
World Competition
See also[edit]
List of environmental economics journals
List of open access journals
List of scholarly journals in international business
List of scientific journals
External links[edit]
Economics journal impact rankings at IDEAS/RePEc
Kodrzycki, Yolanda K.; Yu, Pingkang David (August 2006). "New approaches to ranking economics journals". The B.E. Journal of Economic Analysis & Policy. Walter de Gruyter. 5 (1): 1520–1561. doi:10.1515/1538-0645.1520. SSRN 834564  . PDF
Ritzenberger, Klaus (November 2008). "A ranking of journals in economics and related fields". German Economic Review. 9 (4): 402–430. doi:10.1111/j.1468-0475.2008.00447.x. PDF
This literature-related list is incomplete; you can help by expanding it.
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_economics_journals&oldid=814558421"
Categories:
Economics journals
Economics lists
Lists of academic journals
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Salve Regina
From Wikipedia, the free encyclopedia
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For the university, see Salve Regina University.
Madonna by Raphael, an example of Marian art
Salve Regina of Hermann von Reichenau sung by Les Petits Chanteurs de Passy (Gregorian notation below)
The Salve Regina (/ˌsælveɪ rəˈdʒiːnə/; Ecclesiastical Latin: [ˈsalve reˈdʒina], meaning "Hail Queen"), also known as the Hail Holy Queen, is a Marian hymn and one of four Marian antiphons sung at different seasons within the Christian liturgical calendar of the Catholic Church. The Salve Regina is traditionally sung at Compline in the time from the Saturday before Trinity Sunday until the Friday before the first Sunday of Advent. The Hail Holy Queen is also the final prayer of the Rosary.
The work was composed during the Middle Ages and originally appeared in Latin, the prevalent language of Western Christianity until modern times. Though traditionally ascribed to the eleventh-century German monk Hermann of Reichenau, it is regarded as anonymous by most musicologists.[1] Traditionally it has been sung in Latin, though many translations exist. These are often used as spoken prayers.
Contents
[hide]
1 Background and history
2 Latin text
3 Translations
4 English hymns based on the Latin original
5 Musical settings
6 In popular culture
7 See also
8 References
9 External links
Background and history[edit]
Marian antiphons have been sung, since the thirteenth century, at the close of Compline, the last Office of the day. Peter Canisius (d. 1597) noted that one praises God in Mary when one turns to her in song.[2] Liturgically, the Salve Regina is the best known of four prescribed Marian Anthems recited after Compline, and, in some uses, after Lauds or other Hours.[3] Its use after Compline is likely traceable to the monastic practice of intoning it in chapel and chanting it on the way to sleeping quarters.[4]
It was set down in its current form at the Abbey of Cluny in the 12th century, where it was used as a processional hymn on Marian feasts. The Cistercians chanted the Salve Regina daily from 1218.[3] It was popular at medieval universities as evening song, and according to Fr. Juniper Carol, it came to be part of the ritual for the blessing of a ship.[4] While the anthem figured largely in liturgical and in general popular Catholic devotion, it was especially dear to sailors.[5]
In the 18th century, the Salve Regina served as the outline for the classic Roman Catholic Mariology book The Glories of Mary by Alphonsus Liguori. In the first part of the book Alphonsus, a Doctor of the Church, discusses the Salve Regina and explains how God gave Mary to mankind as the "Gate of Heaven".[6]
It was added to the series of prayers said at the end of Low Mass by Pope Leo XIII.[5]
The Salve Regina is traditionally sung at the end of a priest's funeral Mass by the decedent's fellow priests in attendance.
As a prayer, it is commonly said at the end of the rosary.[7]
Latin text[edit]
Madonna by Tiepolo, 1760
Salve, Regina, Mater misericordiæ,
vita, dulcedo, et spes nostra, salve.
Ad te clamamus exsules filii Hevæ,
Ad te suspiramus, gementes et flentes
in hac lacrimarum valle.
Eia, ergo, advocata nostra, illos tuos
misericordes oculos ad nos converte;
Et Jesum, benedictum fructum ventris tui,
nobis post hoc exsilium ostende.
O clemens, O pia, O dulcis Virgo Maria.
In some cases, the following versicle, response, and collect are added:
℣ Ora pro nobis, sancta Dei Genitrix.
℟ Ut digni efficiamur promissionibus Christi.
Oremus.
Omnipotens sempiterne Deus, qui gloriosæ Virginis Matris Mariæ corpus et animam, ut dignum Filii tui habitaculum effici mereretur, Spiritu Sancto cooperante præparasti: da, ut cuius commemoratione lætamur; eius pia intercessione, ab instantibus malis, et a morte perpetua liberemur. Per eundem Christum Dominum nostrum.
℟ Amen.[8]
Translations[edit]
Part of a series on the
Mariology
of the Catholic Church
The Immaculate Conception, by Murillo
Overview[show]
Prayers
Antiphons
Hymns to Mary
Devotional practices
Prayers[show]
Angelus
Fátima Prayers
Flos Carmeli
Hail Mary
Hail Mary of Gold
Immaculata prayer
Magnificat
Mary, Mother of Grace
Mary Our Queen
Memorare
Sub tuum praesidium
Antiphons[show]
Alma Redemptoris Mater
Ave Maris Stella
Ave Regina Caelorum
Salve Regina
Hymns to Mary[show]
Hail Queen of Heaven, the Ocean Star
Immaculate Mary
Lo, How a Rose E'er Blooming
O Sanctissima
Regina Coeli
Stabat Mater
Devotional practices[show]
Acts of Reparation
Consecration to Mary
First Saturdays
Rosary
Seven Joys of the Virgin
Seven Sorrows of Mary
Three Hail Marys
Litany of the Blessed Virgin Mary
Catholicism portal
v
t
e
Variations exist among most translations.
Traditional English Translation (North America):
(in the version used by Catholics in the United Kingdom and in the Anglo-Catholic version, the wording "mourning and weeping in this vale of tears" is used in the 5th line)
Hail, holy Queen, Mother of Mercy,
Our life, our sweetness and our hope.
To thee do we cry,
Poor banished children of Eve;
To thee do we send up our sighs,
Mourning and weeping in this valley of tears.
Turn then, most gracious advocate,
Thine eyes of mercy toward us;
And after this our exile,
Show unto us the blessed fruit of thy womb, Jesus.
O clement, O loving,
O sweet Virgin Mary.
℣ Pray for us, O holy Mother of God,
℟ that we may be made worthy of the promises of Christ.
Let us pray:
Almighty, everlasting God, who by the co-operation of the Holy Spirit didst prepare the body and soul of the glorious Virgin-Mother Mary to become a dwelling-place meet for thy Son: grant that as we rejoice in her commemoration; so by her fervent intercession we may be delivered from present evils and from everlasting death. Through the same Christ our Lord. Amen.
The Salve Regina was one of the Leonine Prayers, in which context the collect at the end was replaced by different text:
Let us pray:
O God, our refuge and our strength, look down with mercy upon the people who cry to Thee; and by the intercession of the glorious and immaculate Virgin Mary, Mother of God, of Saint Joseph her spouse, of the blessed Apostles Peter and Paul, and of all the saints, in Thy mercy and goodness hear our prayers for the conversion of sinners, and for the liberty and exaltation of our Holy Mother the Church. Through the same Christ Our Lord. Amen.
The United States Conference of Catholic Bishops has a more modern translation:
A painting of the prayer inside Sorrowful Mother Shrine Chapel (Bellevue, Ohio)
Hail, holy Queen, mother of mercy,
our life, our sweetness, and our hope.
To you we cry, poor banished children of Eve;
to you we send up our sighs,
mourning and weeping in this valley of tears.
Turn, then, most gracious advocate,
your eyes of mercy toward us;
and after this, our exile,
show unto us the blessed fruit of your womb, Jesus.
O clement, O loving, O sweet Virgin Mary.
V. Pray for us, O holy Mother of God.
R. That we may be made worthy of the promises of Christ.
Let us pray:
O God, whose Only Begotten Son,
by his life, Death, and Resurrection,
has purchased for us the rewards of eternal life,
grant, we beseech thee,
that while meditating on these mysteries
of the most holy Rosary of the Blessed Virgin Mary,
we may imitate what they contain
and obtain what they promise,
through the same Christ our Lord.
Amen.
English hymns based on the Latin original[edit]
The Divine Office offers the following hymn as an alternative to the Latin:
Hail, our Queen and Mother blest!
Joy when all was sadness,
Life and hope you gave mankind,
Mother of our gladness!
Children of the sinful Eve,
Sinless Eve, befriend us,
Exiled in this vale of tears:
Strength and comfort send us!
Pray for us, O Patroness,
Be our consolation!
Lead us home to see your Son,
Jesus, our salvation!
Gracious are you, full of grace,
Loving as none other,
Joy of heaven and joy of earth,
Mary, God's own Mother!
Catholic missals generally list two or three verses similar to the following:
Version generally used in Catholic missals
Hail, Holy Queen enthroned above, O Maria!
Hail, Mother of mercy and of love, O Maria!
Triumph all ye cherubim!
Sing with us ye seraphim!
Heaven and earth resound the hymn!
Salve, salve, salve, Regina!
Our life, our sweetness here below, O Maria!
Our hope in sorrow and in woe, O Maria!
Triumph all ye cherubim!
Sing with us ye seraphim!
Heaven and earth resound the hymn!
Salve, salve, salve, Regina!
And when our last breath leaves us, O Maria!
Show us thy son Christ Jesus, O Maria!
Triumph all ye cherubim!
Sing with us ye seraphim!
Heaven and earth resound the hymn!
Salve, salve, salve, Regina!
As with many hymns, many more verses exist, but are rarely printed or sung. The Latin text[9] from which these verses are translated is:
Salve, Regina coelitum, o Maria!
Sors unica terrigenum, o Maria!
Jubilate, Cherubim,
Exsultate, Seraphim!
Consonate perpetim:
Salve, salve, salve, Regina.
Mater misericordiae, o Maria!
Dulcis parens clementiae, o Maria!
Jubilate, Cherubim,
Exsultate, Seraphim!
Consonate perpetim:
Salve, salve, salve, Regina.
Musical settings[edit]
Main category: Salve Regina settings
As an essential component of the Compline service, the hymn has been set to music by various composers, including Victoria, Palestrina, Josquin and Lassus. Vivaldi, Handel, and Liszt composed their own settings in later years. Schubert composed no less than four versions. Arvo Pärt composed a setting first performed in the Essen Cathedral in 2002. Olivier Latry premiered in 2007 an organ work Salve Regina which reflects in seven movements the lines of the hymn in Gregorian chant.
In popular culture[edit]
In 1976 the words of the first verse of the Salve Regina were used as a repeating theme in the song Oh What a Circus in the musical Evita, with music by Andrew Lloyd Webber and lyrics by Tim Rice.
The 1992 comedy film Sister Act features a performance of the hymn influenced by rock and roll and gospel music.
See also[edit]
Dialogues of the Carmelites
Lumen gentium
Prayer to Mary (Blessed Virgin Mary)
Queen of Heaven
References[edit]
Jump up ^ Lawrence Gushee, "Hermannus Contractus [Hermann der Lahme, Hermann von Reichenau]," in The New Grove Dictionary of Music and Musicians, 2nd edition (2001): "Although the most venerable ascriptions to Hermannus – the Marian antiphons Alma Redemptoris mater and Salve regina – have been taken away by most recent scholarship, the true authorship is still, and possibly will be for ever, the subject of controversy."
Jump up ^ ""Salve Regina", International Marian Research Institute, University of Dayton". Campus.udayton.edu. 2013-02-27. Retrieved 2014-07-10.
^ Jump up to: a b "Frisk, M. Jean. "Marian Antiphons", the Marian Library, University of Dayton". Campus.udayton.edu. Retrieved 2014-07-10.
^ Jump up to: a b "Carol OFM, Juniper. ''Mariology'', vol.3". Campus.udayton.edu. 2013-02-27. Retrieved 2014-07-10.
^ Jump up to: a b Herbermann, Charles, ed. (1913). "Salve Regina". Catholic Encyclopedia. New York: Robert Appleton Company.
Jump up ^ Saint Alphonsus Liguori, The Glories of Mary, Liguori Publications, 2000, ISBN 0-7648-0664-5
Jump up ^ "The Rosary: Salve Regina (Hail, Holy Queen)", Godzdogz, Blackfriars, Oxford, October 2, 2014
Jump up ^ "Salve Regina". Preces-latinae.org. Retrieved 2014-07-10.
Jump up ^ Salve, Regina coelitum, Complete text in The Roman Hymnal, New York 1884, pp. 120–21
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Salve Regina". Catholic Encyclopedia. New York: Robert Appleton.
External links[edit]
Wikimedia Commons has media related to Salve Regina.
Latin lyrics sung by the Benedictine Monks of Santo Domingo de Silos.
"Salve Regina", International Marian Research Institute, University of Dayton
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