In mathematics, summation (denoted by the large Greek sigma symbol) is a set of summands. What is the amount? This is the result of such an action. If numbers are added one after the other from left to right, then the subtotal is a partial sum.
What is the amount?
The numbers to be added can be integer, rational, real or complex. In addition to them, other types of values ββcan 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 summation always gives a clearly defined value. Summing an infinite sequence of values ββis 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 determined by an expression whose value is the sum of the digits included in it, for example 3 + 7 + 2 + 1 = 13. Since the addition is associative, the sum does not depend on how the terms are grouped for example, (3 + 7) + (2+ 1) and 3 + ((7 + 2) + 1) are both nine, so they usually do without parentheses. Addition is also commutative, so the permutation of the terms does not change the value of the sum. It is worth noting that for infinite summation this property may not work.
There are no special notations for summing sequences of this kind. There is only a slight nuance if there are less than two elements. The record of summing up the sequence of one term 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 you can denote its value "0"). If, however, the sequence members are defined by a certain template, for example, by a function, then the summation operator can be useful or even essential.
Record
To understand what the amount is, it is also necessary to parse its appearance.
To summarize a sequence of integers from 1 to 100, an expression that includes an ellipsis is often used to indicate missing members: 1 + 2 + 3 + 4 + ... + 99 + 100. The pattern is pretty easy to see in this example. However, for more complex cases, it is necessary to precisely indicate the rule used to search for the value of the elements, which can be achieved using the summation operator "Ξ£". Using this symbol (sigma), you can apply the following record:
The value of this expression is 5050. It can be found using mathematical induction, where the second part of the formula comes from.
For different sequences, the formula will be modified. The writing process comes down to searching for the prototype of some infinite sequence and then describing it with a formula. Having done this, it is easy to understand what the amount is in a particular case.
When it is necessary to clarify that numbers are added along with their signs (plus or minus), the term algebraic sum is used. For example, in the theory of electrical circuits, the laws of the Kirchhoff circuit consider the algebraic sum of currents in a network of conductors meeting at a point, giving opposite signs to the currents flowing in and out of the node.