Sequence

As a result or sequence is called in mathematics a list of ( family) of finite or infinite number of consecutively numbered objects (such as numbers). The same object can also occur in a sequence more than once. The object with the number, it says here, too: with the index -th element is or -th component called the sequence. Since infinite sequences can not be fully listed, an education law for the followers must be known or demonstrated unequivocally from the written down early members. Finite as infinite sequences are found in all areas of mathematics. With infinite sequences whose elements are numbers, concentrated primarily the analysis.

If the number of members of a finite sequence, then one speaks of a sequence of length, one - membered sequence or of a tuple. The result without limbs, so the index range is empty, empty sequence, 0 -membered sequence or 0 -tuple is called.

• 6.1 monotony 6.1.1 Proof of monotony

• 6.2.1 Proof of boundedness / determining a barrier

• 7.1 Arithmetic Sequences and Series
• 7.2 episodes based on the power function
• 7.3 Geometric consequences

Examples

Spelling

General to write for a sequence, ie at finite sequences and infinite sequences. stands for any follower; the parenthesis combines these into a sequence. Instead of the round brackets are sometimes also used lace ( so ); instead of commas, semicolons can be used if a likelihood of confusion with the decimal point.

The difference from the amount of the followers is, to the order that it arrives, and that a plurality of follower members can have the same value.

Formal definition

Formally defined an infinite sequence as a mapping

To each index from the set of natural numbers used as index set a follower of the target amount assigns. The choice of the initial index is ultimately arbitrary. In school mathematics and in the most common applications is the set of real numbers. However, also be considered as sequences of sets and sequences of functions.

For a finite sequence ( tuple ) with members one defines the index instead of from a finite set, usually from either the quantity or the quantity. Occasionally can be found for such index sets the notation.

Applications

Time series, as they arise, for example, by recording temperature observations or economic data can be mathematically interpreted as consequences.

In computer science, one can fields ( arrays) regarded as finite sequences.

Infinite sequences can converge to a limit. The theory of limits of infinite sequences is an important basis for the analysis, because it is based on the calculation of limits of functions, the definition of the derivative ( the derivative as a limit of a sequence of difference quotients ) and the Riemann integral term. Important consequences are obtained as coefficients of Taylor series of analytic functions. Some elementary functions perform here on special effects, the tangent function on the Bernoulli's or the hyperbolic secant of the Eulerian numbers. To prove the convergence of a sequence, the method of Full induction is a useful tool.

A number is a special sequence of numbers, the TES member resulting from the sum of the first links of a different sequence of numbers. For example, there is the row ( 1, 3, 6, 10, 15, ... ) of the sequence (1, 2, 3, 4, 5, ...). Rows are used in many areas of mathematics. See article series (mathematics).

Education law of a sequence

There are several ways to specify a sequence:

• Calling all followers (only for finite sequences )
• Function
• Series
• Recursion
• Algorithm

A finite sequence can specify one by one calls all followers. For an infinite sequence will not do, you have to notify the Education Act the result in another form instead.

Sequences whose education law can be notified as function rule or recursion are sometimes called regular sequences. This is not a mathematically rigorous classification since, after each episode, which ever is well-defined, has a function rule. In irregular consequences can only say that the specification of a function rule according to current mathematical knowledge is complex and / or via algorithmic rules that can not be represented in functional form with the usual notation. This issue is related to the difficult question of predictability.

Indication of early members

The set in some intelligence tests task to continue a sequence whose first members are given, is problematic from a mathematical point of view. Also, by so many early members of the further course of a sequence is not clearly defined. There are only more or less plausible sequels.

• Where is 0, 1, 2, 3 most plausible is the continuation of 4, 5, 6, ..., ie the sequence of all natural numbers. It is also possible to continue 0, 1, 2, 3, 0, ..., as a periodic sequence of the smallest positive residue of natural numbers modulo 4 In a computer, integers are often 32-bit two's complement so as to absolute smallest residues modulo 232 shown. When successive increase of a register one goes through then the numbers 0, 1, 2, 3, ..., 2147483647, -2147483648, -2147483647, ..., -1, and periodically.
• For the numbers 3, 1, 4, 1, 5 is a plausible continuation of 1, 6, 1, 7, ... Other the decimal representation of the number circle would recognize and ... propose the continuation of 9, 2, 6.

The Online Encyclopedia of Integer Sequences ( OEIS ) contains thousands of mathematically relevant consequences. In it you can search for a given subsequence.

Specifying a function rule

For many, but not all consequences can be the function rule

Specify as a closed equation.

In the following examples we attach indices from the set based on:

• The sequence of natural numbers 0, 1, 2, 3, ... This example is especially because the values ​​of index match and follow-up member. The function rule is simply

• The sequence of the odd numbers 1, 3, 5, 7, ... has the function of provision

• The sequence of powers of 1, 2, 4, 8, ...

It CONTINUED tasks

The problem of determining for a given function rule, the initial members, is easy to solve. It takes successively the values ​​, etc., puts them each a in the function rule and calculated in this way, the followers, etc. The purpose of this bill is to make a first impression on the course of a sequence. But beware: A sequence can be used for really large indexes take a very different course than was to be expected after the first ten or a hundred members. For example, the sequence increases to monotonically, but then decreases again, as can be verified by inserting orders of magnitude higher.

The inverse problem of determining for given initial members of a function rule, however, is much more difficult. Strictly speaking, there can be no unique solution, since the beginning of each episode can be continued as described above in various ways. In practice, this task is therefore provided only for sequences whose members, etc. depend to some extent a manageable way from the index. Specifically, the following properties can be verified:

• If the sequence alternating? If so, you get the correct sign by a factor in the function rule. Example: 0, -1, 2, -3, 4, ..., the provision.
• Are the followers fractures? If so, construct one independently operating rules for the numerator and denominator. For example: 1/1, 2/ 2, 3/ 4, 4/ 8, ..., the provision.
• Take the sequence terms to constant differences to (or from, with )? If so, you have an arithmetic sequence. Example: 1, 3, 5, 7, ..., the provision.
• Do the differences between consecutive terms a simpler Education Act as the followers themselves? If so, can you interpret the result as a series (see below). For example, for 1, 3, 6, 10, 15, ... are the differences 1, 2, 3, 4, ...
• Standing consecutive sequence elements to each other in a constant ratio? If so, you have a geometric sequence. Example: The sequence 100; 80; 64; 51.2; ... Decreases from term to term by a factor of 0.8; So is the rule.

Is it difficult to find a function rule in that the first one or two sequence elements often seem to fall out of the frame ( with the indices 0 and 1). The reason is that a summand 0, 1 or a factor exponent 0 or 1 is not advertised as a rule, but immediately be calculated. In the shortened form 1, 1, 3/ 4, 1/ 2, ... is the above example, 1/1, 2/ 2, 3/ 4, 4/ 8, ..., the function rule difficult to see.

Specified as a series

A result, the -th element can be written as a sum of the first links of a different sequence, i.e., a series of:

The written using the summation sign expression is thus an abbreviation for the expression. Inside and outside of the sum sign different indices can be used. That specifically and were chosen corresponds to a widespread convention, but is not mandatory.

To calculate the actual value, a specific numerical value for the index must be specified. In contrast, the index is not vorzugebender ( from the outside) value, but determined by the summation rule itself. Whatever is given for the " run index " must successively the values ​​0, 1, ..., used, and the sum will be the corresponding, , ..., is calculated.

One can interpret each sequence as a series by successively from the differences between the following members a corresponding sequence

Constructed. Sequence and series are not so sharply separated from each other. The time series of economists are actually consequences. Many explanatory models model not absolute values, but their changes over time, which speaks as members of a series for the conception of the absolute values ​​.

Concrete benefits does the interpretation of a sequence as a series, if you can perform the summation for any. Summation formulas are known for example for arithmetic series and geometric series.

The interpretation of an infinite sequence as a series makes it easier to determine whether, and if so, against which limit the sequence converges. For infinite series has its own convergence criteria. Conversely, it is from the convergence of a series (that is, in the notation described above, the convergence of ) always indicates that the result of the " summands" (ie the sequence in the above notation) converges to zero.

Specifying a recursion

The Education Act of a sequence can also be specified recursively. These are called initial values ​​( with, is mostly or ) as well as a provision, as a follower of the previous links can be calculated.

The best-known example of a sequence that is much easier to describe by a recursion than by a function rule, is the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, ... For them, are given two initial members and as well as the recursion

The explicit formula of Moivre and Binet for the followers

Is closely related to the Golden Section and the Golden Number. Note that all are integers, because the odd powers cancel out of.

Some consequences can be derived a recursion reversed from the function rule. For example follows for the geometric sequence of the function rule

The recursion

The recursion

Defines the sequence of rational numbers 2, 3/2, 17/ 12, ..., which converges to.

Indication of an algorithm

For some episodes there is a clearly defined design procedure ( algorithm ), but no functional rule. The best known example is the sequence of prime numbers 2, 3, 5, 7, 11, ... Already the ancient Greeks (and possibly Indians ) it was known how to calculate always more terms of this sequence. One possibility is to use the Sieve of Eratosthenes. However, there is no method to specify a given the - th prime number without first compute the entire sequence from the first to the costs prime number (or look up ). If you want to know not the tenth or the hundredth, but the -th prime number, this increases the computational complexity greatly.

The length of the shortest algorithm generates a sequence, ie their Kolmogorov complexity (sometimes this term is used in a narrow sense on strings, that is finite sequences of finite amounts of target used). Although it depends on the programming language used; after Invarianztheorem however, the lengths for different languages ​​differ by only a only language-dependent additive constant.

Characterization of sequences

What functions can also characterize sequences of numbers on their pitch behavior and image area:

Monotony

A sequence is called monotone increasing if it remains the same or increases, so if out applies to all from member to member. The sequence is called strictly monotonically increasing if it increases from member to member, if so applies from for all. The terms monotonically decreasing and strictly decreasing are defined analogously.

Proof of monotony

Suspected you that a sequence is not monotone (or strictly ) is just put a couple of indices in the function rule, calculates the associated followers, hoping to find a counterexample. Example: by given sequence is not monotonic, but because.

Example, if one suspects that a sequence falls monotonically, to write, evaluates on both sides of the function rule from ( by clicking on the right side instead of using the rule ), and analyzed the resulting inequality by being simplified by equivalence transformations. Example: lists, which is equivalent to or for the true statement.

Some operating rules can be broken down by term transformations in a sum of constant terms and a known, simpler result, the slope behavior is already known. For example. If you know that strictly falls, one can conclude that a strictly monotonic increasing. Because of Term 2 is constant, increases monotonically.

Narrowness

A sequence is called bounded above if it has an upper bound, so that from true for all. The least upper bound of a sequence is also called its supremum. The terms bounded below, greatest lower bound and infimum are defined analogously. A sequence which is bounded above and below, is called bounded sequence.

Proof of boundedness / determining a barrier

Proof by counter-example is not possible, because no matter how many examples can not ensure that there are not some very large or very small number by which the sequence is bounded one.

It must be assumed, therefore, that there is a barrier. Now the right inequality is recognized, that is, for an upper bound so. On the left side of the inequality, the function rule is applied, and then dissolved. This gives ( with some luck ) a result of the form or, where is one of dependent term. In the first case, it was found that the sequence is not bounded above (no matter how large, it is always possible to find an even bigger that violates the inequality). In the second case, one tries to find one that is for. For this is always true, and thus succeeded in establishing that an upper bound is.

Again, the proof can easily make if it is possible to disassemble the function rule into a sum of simpler terms.

Other

• A result, the values ​​of which are alternately positive and negative states alternately.
• A sequence whose members all agree, constant sequence is called.
• A sequence whose members all agree on a certain element, stationary sequence is called
• A sequence that converges to 0, is called a null sequence.
• A sequence consisting of iterations of a finite subsequence is, periodically. There is a period length, and is made ​​for all. Subsequence is to be understood here as a result of the chosen quantity.

An interesting task from the analysis is to determine whether a sequence converges, and in the case of convergence, against which limit. An infinite sequence that does not converge, may nonetheless possess accumulation points (example: the sequence -1 / 2, 3/4, -5 / 6, 7/8, ... has the accumulation points -1 and 1). In particular, every bounded sequence in the set of real numbers at least one accumulation point ( theorem of Bolzano - Weierstrass ).

The above characterization of a row over her pitch behavior and image area may help to determine whether and, if against which limit it converges. Especially useful here is the monotonicity criterion by which a monotonically increasing, bounded above sequence in the set of real numbers always converges, with its limit with their supremum matches (example: the sequence 0, 1/ 2, 2/ 3, 3 / 4, ... converges to its supremum 1). According converges a monotonically decreasing, bounded below a row against their infimum.

The characterization criteria monotonicity and boundedness can be generalized for all sequences whose target quantity is ordered. Constant, stationary and periodic sequences can be used for any destination areas that define convergent sequences for an arbitrary metric space as the target area.

Important consequences

Most of the known sequences of numbers can be looked up in the On- Line Encyclopedia of Integer Sequences ( OEIS ) by Neil Sloane. This database contained in February 2009 over 155,000 descriptions of numerical sequences.

Arithmetic Sequences and Series

An arithmetic sequence is a sequence with a constant difference between adjacent elements. Examples are the results of frequently-used even numbers 2, 4, 6, ... to the function provision

And the odd numbers with the function rule

General is the function rule

The constant difference referred.

Consequences that can be attributed to arithmetic sequences are called arithmetic sequences of higher order. So the sequence of triangular numbers is an arithmetic sequence 2nd order.

Arithmetic series -th order are precisely those consequences, which can be described by a polynomial of degree. This polynomial can be found by Lagrange interpolation from any followers. The triangular numbers obey example the Education Act.

Consequences based on the power function

A power sequence is a sequence for which the power function which provides members ( generating function )

The sequence of square numbers: 0, 1, 4, 9, ... has the function of provision. The sequence of square numbers is also an arithmetic progression of 2nd order, since it can be conceived as a series, which is based on the sequence of odd numbers.

The sequence of the cubic numbers 0, 1, 8, 27, ... has the provision

What you can for -th powers of natural numbers

Can be generalized, with an arbitrary real number may be. Is obtained with the result of the square roots of integers,

With negative exponents is to be noted that not exist. For example, it is not possible, with the function and regulation

To calculate. One can exclude the index 0, so are limited to the index set. Often, however, it is more appropriate to let the index set unchanged and instead the function rule in

Amend. Then are the first terms of the sequence 1, 1/ 2, 1/ 3, 1/ 4, ... In the same way you can place a function rule for any exponent:

Geometric consequences

Just as in an arithmetic sequence successive links have a constant difference, so stand in a geometric sequence

Consecutive links in a constant ratio to each other. For example arises with and the sequence of powers of two

Thus, for example, for the first ten elements, the sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ( each element is twice as large as the previous one ). Importantly this episode especially for the conversion of those used in the computer science binary numbers to decimal (and vice versa). A geometric progression with converges to zero, such as the sequence 1; 0.1; 0.01; To ...:

When one receives the trivial sequence 1, 1, 1, ...; when, is obtained from

The fundamental alternating sequence 1, -1, 1, -1, ...

An example of the daily application of the geometric progression is the equal temperament of the musical scale - the successive terms, here semitones, have a frequency ratio constant to each other.

Generalizations

In the topology of a network is a generalization of a sequence.

As with functions can be defined in addition to the consequences specified here with values ​​in quantities and sequences with values ​​in a real class, so for example, sequences of sets or groups.

Sequence spaces

From Follow the sequence spaces can be formed, which are used mainly in the functional analysis for the construction of examples.