Series (mathematics)
A number, especially in older representations also called infinite series, is an object from the mathematical subfield of Analysis. Clearly, a number, a sum with infinitely many summands. Precisely a series is defined as a sequence whose elements are the partial sums of another sequence. The - th partial sum is the sum of the first (from the infinite number of ) summands. If the sequence has a limit of these partial sums, so that the value or the sum of the series is known.
- 8.1 Examples
- 9.1 Representation of mathematical constants
- 9.2 rows of functions 9.2.1 Power series
- 9.2.2 Fourier series
- 9.2.3 Dirichlet series
Definition
If any sequence given you a new episode can out of it with
. construct These members of the sequence are called ( th ) partial sums. The result of these links, so the sequence of partial sums is called -th row. If the series converges, it is called its limit
Value of the series or sum of the series.
Notation
For rows there are different notations depending on the context. In this article, be used as indices for the elements of sequence and series of the natural numbers including zero. In some applications it is appropriate to start the summation only at index 1, 2 or higher rarely also negative indices (see Laurent series ). Using the summation sign, the individual members of the series can also be abbreviated as
Be written. Likewise, one proceeds in the sequence of the individual members and writes short
Frequently, a part or all of the indexes can be omitted when misunderstandings are excluded. Is like it is here in the context of calculations clearly with infinite series that is generally begun to number at 0, as is
Evaluation and classification
If and are thus defined for all nonnegative indices i and n, an infinite series is thus possible to form: when the limit of the sequence of partial sums
Exists, they say, the series converges; the limit value S is called the total number or value of the row. Using the summation sign this sum can also be abbreviated as
Be written.
A series is called divergent or limit their non-existent, if the series does not converge. You say definitely divergent or improperly convergent if the partial sums to - ∞ or ∞ aspire. Otherwise, the series is called indefinitely divergent; while they may have accumulation points or not.
With different convergence criteria can be used to determine whether a series converges.
Examples
For some simple finite series, one can calculate the sum explicitly, for example, for arithmetic series such as
Or the sum of squares is first n
Or the series
A proof of such formulas can be done eg via induction. However, there are constructive ways to add rows explicitly: Euler's summation formula, telescopic sums, summation and rearrangement known series. Further such summation formulas can be found in the formulary arithmetic.
Another classic series is the geometric series, the name is derived from the geometric progression ( for ). The infinite geometric series is:
Additional examples of finite series can be found in the article sum.
A special geometric series is
This notation referred to by the above given representation of the limit of the sequence
One can visualize the convergence of this series on the number line: Consider a line of length two in front, on the successive sections with lengths of 1, 1 /2, 1 /4, etc. are highlighted. There are on this line is still room for another section, since there is still so much space as the last section was long: When we have marked the route 1/2, we have a total of 3/2 consumed, so there are still 1/2 left. If we now 1/4 prune, remains another 1/4 left, etc. Since the " remnant " is arbitrarily small, the limit is equal to 2
Convergent geometric series are also an object of the paradoxes of Zeno.
An example of a divergent series with multiple cluster points is the sum of the sequence 1, -1, 1, -1, ... The series alternates between the values 1 and 0 ( the result, however, changes between 1 and -1 ).
Semantics and comparison
The symbol
Come here to two different roles that need to be taken out of the context. Once it symbolizes the value of the series, which exists in the case of convergent series or in the case of divergent series does not exist. On the other hand, it represents the series as a result of the partial sums, regardless of the convergence behavior.
The difference is clear when distinguishing between value- wise and member -wise equality of rows: Two rows are called term-wise equal (or identical or term- consistent ) if their members coincide with the same index. As a symbol of the identity symbol will be used.
The identity of rows, it is irrelevant whether or not to converge. Not so with the "simple " equality: Two rows are called (value of) the same ( or equal in value or sum equal ) if both converge and agree in their values. For this purpose one uses the ordinary equals sign. In character:
The page on the right end ( with the symbol), which contains no limit expression, symbolized here is that a numerical limit exists.
For the geometric series
Applies according to the formula for the value of geometric series:
But:
Calculating with rows
Unlike ordinary (finite) sums, some common rules of addition apply only conditionally for rows. So you can not or only under certain conditions reckon with them as finite sum of expressions.
Sums and multiples
You can add, subtract, or by a fixed factor ( no other number ) multiply ( multiply ) convergent series term by term. The resulting lines are also convergent and its limit is the sum or difference of the limits of the output rows or the multiple of the limit of the output range. That is,
Products
You can absolutely convergent series term by term, multiply each other. The product range is also absolutely convergent and its limit is the product of the limits of the output rows. That is,
Since the spelling ( on the left side of the equation ) of the product range with two indices in certain contexts is " unwieldy ", the product range is also written in the form of Cauchyprodukts. The name arises from the fact that the members of the product line are made with the help of the Cauchy diagonal process, while the members of the output sequences are arranged in pairs in a quadratic scheme and the ( numbered ) diagonals of this scheme are the product links. For the product range you only need then a single index. The product line has then the following form:
Numeracy within the series
Bracketing ( associativity )
It is within a row limbs arbitrarily summarized by brackets. We can add any number of brackets in the " infinite sum expression ," they must not only within one ( of several terms combined) set member. The value of the row is not changed by the additional Beklammerung then.
The other way but you can omit the parentheses without further notice. It can, however, whenever the resultant number is again converging. In this case, also the number is unchanged. If the " underprivileged clinging " series namely is convergent, one can add their same clips again, that had previously been taken away, and the equality of the limit is given by the above, if it confuses the roles and the " underprivileged clinging " series now as range considered, the adding brackets.
Rearrangement ( commutativity )
A rearrangement of a series is represented by a permutation of its index set. If the index set, for example ( as usually), ie the set of natural numbers and a bijective mapping ( permutation ) between the natural numbers as
A rearrangement of the series
You can convergent series while retaining its value then and only then rearrange any if they are necessarily or absolutely convergent. It arises for sure (or absolutely ) convergent series:
Conditionally convergent series may only be finally be rearranged, ie from a certain index must apply for the rearrangement. Infinity rearrangements can change the limit of a conditionally convergent series or let them diverge. You can even let go by rearranging every conditionally convergent series against any limit ( Riemann rearrangement theorem ).
Absolute and unconditional convergence
A number is called absolutely convergent if the series converges its absolute links.
A convergent series is absolutely convergent defined formally as if each of its rearrangements converge again and has the same limit. However, the last property does not need to be provided as each row, all of whose rearrangements are convergent, and each rearranging the same value. A convergent series, which is not necessarily convergent means limited convergent.
In finite- dimensional spaces of the theorem:
For a conditionally convergent series can specify any number and then find a rearrangement of this series converges to exactly this number ( Riemannian rearrangement theorem ). In particular, you can specify as number and not a number, to say that the series should diverge, and finds a suitable rearrangement that does that.
Convergence criteria
In the following, the numbers are always real or complex numbers, and the series S defined as
To prove the convergence of this series, there are several convergence criteria, partly showing the contingent, in part, the greater absolute convergence ( convergence of the series of values of the terms ):
If the series S converges, then the sequence converges () the summands for 0 against Formulated: If () not null sequence diverges the corresponding row. The converse is not universally valid ( a counter-example is the harmonic series ).
If all the members of S are non-negative real numbers, and S converges for all n
Applies with real or complex numbers, then also converges, the series
Absolute, and it is | T | ≤ S.
If all the members are on the S series are non-negative real numbers, S diverges and for all n
Applies with non-negative real numbers, then diverges, the series
If a constant C <1 and an index exists N such that n ≥ N holds for all
Then the series converges absolutely S.
If a constant C <1 and an index exists N such that n ≥ N holds for all
Then the series converges absolutely S.
Is a non-negative monotone decreasing function with
Then S converges if and only if the integral
Exists.
A series of the form
With non-negative on alternating series is called. Such a series converges, when the sequence converges monotonically approaches 0. The converse is not generally valid.
Examples
- A geometric series converges if and only if.
- The Dirichlet series converges if and diverges for what can be shown with the integral criterion. As a function of r considered, this series gives the Riemann zeta function.
- The telescoping series converges if and only if the sequence converges to a number L. The value of the series is then.
Applications
Representations of mathematical constants
In addition to the convergence and the numerical value of a number is also the symbolic value of a series of significance. For example, mathematical constants can thus represent and calculate numerically. Example of ( natural logarithm)
For important series representations tabulations exist in series panels.
Series of functions
Instead of sequences of numbers can also consider sequences of functions and define corresponding rows. Here comes the question of the convergence nor the added according to the properties of the limit function. Conversely, one may ask, by which series to a function can be represented. Such a representation is called expansion.
Power series
Some important functions can be represented as a Taylor series. These are certain infinite series, that contain powers of an independent variable. Such rows are commonly called power series. Will also negative powers of the variable approved, it is called a Laurent series.
Fourier series
As a Fourier series of a function is called its development by trigonometric functions and. The Euler number is also of this type.
Dirichlet series
As a Dirichlet series is called a development
An important example is the series representation of the Riemann zeta function