Summation

A sum is the result of a mathematical addition. In the simplest case, is a sum that is a number which is produced by adding two or more numbers. This term has many generalizations. So you spoke earlier example of summable functions, referring integrable functions.

Word history and meanings

The word was borrowed sum summa in Middle High German from Latin. Summa was to the 19th century in addition to the sum in use and goes on summus back to superus one of the Latin superlative " located above, the / the / the Higher / Upper ", which consequently " of / / the Supreme / Supreme ". imply " The Supreme " because the Romans used to record the total in the top row, so on the summands, and not as common today, " the bottom line ".

In a broader sense refers to a sum vulgarity or epitome.

In everyday language, the sum referred to an amount of money, regardless of whether he has come into existence by the addition or not.

Sum as a result of an addition

In the mathematical expression

The names of the numbers 2 and 3 summands. The entire term is referred to as the "sum of 2 and 3 '.

It may form a sheet of more than two terms, such as 4 7 1. Due to the associativity of addition while no need to specify the order in which the additions are to be executed. How true that is, and the sum can also be written without parentheses.

Due to the commutative law of addition and the order of the summands is irrelevant, that is, it is, for example,

If times the same number is added, then the sum can also be written as a product.

Weighted sum

In some cases, the individual summands are not simply added, but previously multiplied by a weight:

For example

In this case one speaks of a weighted sum. Dividing the weighted sum by the sum of the weights, we obtain the weighted arithmetic mean.

Sum of a sequence number

If a sum has many summands, it is appropriate to agree an abbreviated notation. The sum of the first 100, for example, as natural numbers

Be specified, because it is easy to guess which addends were replaced by the ellipsis.

Just as one goes into the elementary arithmetic of numbers to letters such as invoices bills like, so you can for example generalize the sum of one hundred specific numbers to the sum of any number of arbitrary numbers. For that, a variable is chosen, for example, which indicates the number of summands. In the above case, the sum of the first one hundred natural numbers would. Since arbitrarily large to be allowed, it is not possible to designate all summands by different letters. Instead, a single letter, for example, selected and added to an index. This index successively takes the values ​​1, 2, ... to. The summands can be called. The summands thus form a number sequence (see sequence ( mathematics) ).

We can now for any natural numbers, the sum of the first links of the sequence of numbers as

. Write If, ... used for different values ​​of 1, 2, also form the part, a consequence. Such a sequence of partial sums of the first members of a sequence is called a row.

Example: For the sequence of square numbers. The general rule

The series of partial sums of this sequence begins with, ,. A summation formula now states for arbitrary

Further summation formulas such as The Little Gaussian

Can be found in the formulary arithmetic. The proof of these formulas is via induction.

Notation with the summation sign

Sums over finite or infinite series may be listed instead of ellipsis with the summation sign:

The total mark consists of the great Greek letters Σ (Sigma ), followed by a sequential member (here) is denoted by a previously used index. This index is often referred to as a running index or summation variable or running or count variable. For this purpose, usually the letters, and used. If not clear which variable is the count variable, this must be noted in the text.

What values ​​can take the control variable, and the top is at the bottom, where appropriate, the displayed. There are two options:

This information may be reduced or omitted if it can be assumed that the reader is able to supplement out of context. Of this, use is made in certain contexts in detail: In the tensor calculus is often agreed the Einstein summation convention, according to which even the summation sign can be omitted because the context is clear, that is to sum over all doubly occurring indices. Here is an animation of the sigma notation:

Formal definition

Be a (index ) number, a commutative monoid. For each was a given. Then you can at least be defined for finite index sets by recursion: you bet

And otherwise

Choice of an arbitrary element. Commutativity and associativity of addition in guarantee that this is well defined.

The spelling in this sense is just an abbreviation for with.

If is infinite, is generally defined only if many valid for all but finitely. In this case, is

On the right is by assumption a finite index set, ie, as defined above sum. Are infinite number other than 0, it is in spite of similar spelling no longer a sum, but a number (see below).

Clip conventions and rules of calculation

If the follower as the sum (or difference) communicated, it must be written in brackets:

If the follower as a product (or quotient ) reported, the clip is superfluous:

Caution: In general,

Special sums

For the sum of one summand of:

For one has a so-called empty sum, which is equal to 0, since the index set is empty:

Is the follower constant (more precisely, independent of the control variables ), the sum can be rewritten as a simple product ( if any):

Double sums

Also about sums can be accumulated again. This is especially useful if the first, which in turn contains "internal" sum, an index of "external" as a running index for the sum can be used. One writes, for example:

The following rule applies:

In mathematical physics, also applies to double sums following convention:

An apostrophe on the summation sign means that in the summation summands be skipped, for the match, the two control variables:

Series

If infinitely many terms are summed, so for example

With ( countably ) infinitely many summands different from zero, methods of analysis need to be applied to the corresponding limit

To see if it exists. Such a sum is called an infinite series. As an upper limit to write the symbol for infinity.

Important differences between rows and real totals are for example:

• Is not defined for any (ie convergent ).
• Convergence and value may depend on the order of the summands.
• The interchange of double sums can not always be transferred to rows (double).
• Lack of seclusion: For example, is irrational, even though all the summands are rational.

It is however to be noted that not every sum that has as an upper limit must be an infinite sum. For example, the sum of

For prime numbers and the integer function, namely infinite sum, but only finitely many are nonzero. ( This sum indicates how often the factor in the prime factorization of n occurrences! . )

Related terms

• The disjoint union of sets has a certain formal similarity with the addition of numbers; for example, finite sets and so is the number of elements equal to the sum of the numbers of element and. The Cartesian product is distributive over this summation:

• The analog of categorical view design for vector spaces or Abelian groups is called a direct sum; generally it is called a coproduct.
• A telescoping sum is in mathematics a finite sum of differences, cancel at any two neighboring elements (except the first and the last ) to each other.
• A Pythagorean summation refers to a similar calculation of the addition operation in which the square root is calculated from the sum of the squares of several sizes.