In mathematics, summation (denoted by the large Greek sigma symbol) is a set of summands of numbers. What is the amount? This is the result of such an action. If the numbers are added one after the other from left to right, then the intermediate result is a partial sum.
What is the amount?
The numbers to be summed can be integer, rational, real or complex. In addition to them, other types of values can be added: vectors, matrices, polynomials, and in general elements of any additive group (or even a monoid).
If the number of elements of the terms is finite, then the summation always gives a well-defined value. The summation of an infinite sequence of values is called a series. Its value can often be determined using a limit (although sometimes the value can be infinite).
Sequences
The summation of numbers [3, 7, 2, 1] can be defined by an expression whose value is the sum of the digits included in it, for example 3 + 7 + 2 + 1=13. Since the additionassociatively, the sum does not depend on how the terms are grouped, for example, (3 + 7) + (2 + 1) and 3 + ((7 + 2) + 1) both equal nine, so parentheses are usually dispensed with. Addition is also commutative, so rearranging the terms does not change the value of the sum. Note that this property may not work for infinite summation.
There is no special notation for summing sequences of this kind. There is only a slight nuance if there are less than two elements. The summation of a sequence of one member does not contain a plus sign (it is indistinguishable from the form of the number itself), and if there are no elements at all, then it cannot even be written (but instead it can be denoted by its value "0"). If, however, the terms of the sequence are specified by a specific pattern, such as a function, then the summation operator can be useful or even essential.
Record
To understand what a sum is, it is also necessary to parse its appearance.
To sum a sequence of integers from 1 to 100, an expression is often used that includes an ellipsis to indicate the missing members: 1 + 2 + 3 + 4 + … + 99 + 100. The pattern is fairly easy to see in this example. However, for more complex options, it is necessary to specify exactly the rule used to find the value of the elements, which can be achieved using the "Σ" summation operator. Using this symbol (sigma), you can apply the following notation:
The value of this expression is 5050. It can be found using mathematical induction, which is where the second part of the formula came from.
For different sequences, the formula will change. The recording process is reduced to searching for the pre-image of some infinite sequence and then describing it with a formula. Having done this, it is not difficult to understand what the amount is in a particular case.
When it is necessary to clarify that numbers are added together with their signs (plus or minus), the term algebraic sum is used. For example, in electrical circuit theory, Kirchhoff's circuit laws consider the algebraic sum of the currents in a network of conductors meeting at a point, giving opposite signs to the currents flowing in and out of a node.
