What Is Summation Properties In Math With Real Examples

Summation properties will help you understand it well.What is a summation , how do you use the summation symbol and what parts does it consist of? Could you give some examples of calculating a summation and show me the properties it enjoys?

What Is Summation Properties In Math With Real Examples

The summation allows you to compactly write the sum of a finite or infinite number of terms.The summation symbol is the capital sigma letter Σ but written alone it has no meaning; as shown in the following image, a summation is made up of several parts, all essential to be able to calculate the summation.

summation properties

As we can see, in addition to the summation symbol we need:

– a letter, called the summation index , used to define the algebraic expression; the letters most commonly used for the summation index are k, i and j.

– An algebraic expression that depends on the summation index which tells us how the terms to be added are made.

– Two natural numbers n and m which express the range of values ​​between which the index varies and therefore tell us how many and which are the addends .

As we will see shortly in detail, often the natural number m is replaced by the symbol of infinity or, again, we may find ourselves faced with even more general situations in which the summation index is not limited to varying between two integers.

Examples of Summation Properties

In order not to leave room for doubt, let’s see some examples of summation and then explain how to concretely calculate a summation.

\ sum_ {k = 0} ^ {11} k, \ quad \ sum_ {k = 3} ^ {8} [k-2], \ quad \ sum_ {k | 12} k ^ 2, \ quad \ sum_ { k \ in \ {4,16,36 \}} \ sqrt {k}, \ quad \ sum_ {k = 1} ^ {\ infty} \ left (\ frac {1} {2} \ right) ^ k

all are examples of summations.

Calculation of a summation

Now that it is clear what it looks like, let’s see how a summation is calculated by examining the previous examples.

With the exception of summations whose symbol at the top is ∞, to calculate a summation, the integer values ​​of the interval between which the index varies must be substituted in the algebraic expression, and then the terms obtained are added together. Proceeding in this way we will have:

\ sum_ {k = 0} ^ {11} k = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66

Since the summation index k varies between 0 and 11, in the algebraic expression we have substituted for k all the natural numbers between 0 and 11 thus obtaining the various terms to be added.

\ begin {align *} \ sum_ {k = 3} ^ {8} [k-2] & = (3-2) + (4-2) + (5-2) + (6-2) + (7 -2) + (8-2) = \\ & = 1 + 2 + 3 + 4 + 5 + 6 = 21 \ end {align *}

The various addends have been obtained by substituting integers between 3 and 8 instead of ki, which are the values ​​between which the index k varies.

\ sum_ {k | 12} k ^ 2 = 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + 4 ^ 2 + 6 ^ 2 = 1 + 4 + 9 + 16 + 36 = 66

The writing k | 12 is a compact way of indicating that k must divide 12; since the divisors of 12 are 1, 2, 3, 4 and 6, to calculate the previous summation in the algebraic expression (k 2 ) we have substituted these values.

\ sum_ {k \ in \ {4,16,36 \}} \ sqrt {k} = \ sqrt {4} + \ sqrt {16} + \ sqrt {36} = 2 + 4 + 6 = 12

In this case, the summation index belongs to a set well-defined, and then to calculate the sum we replaced the elements of the place of k.

Infinite summations

When the infinity symbol is found above the summation symbol, we are faced with a sum of an infinite number of terms. To calculate these sums it would be impossible to proceed as done above, ie by direct substitution of the index values, as these values ​​are infinite.

These summations are called numerical series and an entire part of the university course of Analysis 1 is dedicated to them ; for those wishing to know more, refer to the page of the homonymous link.

Properties of the summation

The summation, understood as the sum of a certain number of addends, has various properties that often facilitate the calculation of the sum and which are listed below.

Associative property : if we are in the presence of two or more summations in which the indices have the same definition interval, then the algebraic sum of the summations is equal to the sum of the algebraic sum.

\ sum_ {k = n} ^ {m} f (k) \ pm \ sum_ {k = n} ^ {m} g (k) = \ sum_ {k = n} ^ {m} [f (k) \ pm g (k)]

Dissociative property : the sum of an algebraic sum is equivalent to the algebraic sum of the single summations.

\ sum_ {k = n} ^ {m} [f (k) \ pm g (k)] = \ sum_ {k = n} ^ {m} f (k) \ pm \ sum_ {k = n} ^ { m} g (k)

Distributive property : A factor that does not depend on the index can be extracted from the summation or, equivalently, a factor external to the summation can be brought into it.

\ alpha \ cdot \ sum_ {k = n} ^ {m} f (k) = \ sum_ {k = n} ^ {m} [\ alpha \ cdot f (k)]

Decomposition of indices : a summation can be decomposed into the sum of two or more summations by appropriately varying the interval of definition of the index.

\ sum_ {k = n} ^ {m_1 + m_2} f (k) = \ sum_ {k = n} ^ {m_1} f (k) + \ sum_ {k = m_1 + 1} ^ {m_1 + m_2} f (k)

Translation of the indices : the definition interval of the index can be changed at will, provided, however, that the algebraic expression of the summation is changed accordingly.

\ sum_ {k = n} ^ {m} f (k) = \ sum_ {k = n + \ ell} ^ {m + \ ell} f (k- \ ell)

Summations in Word, Excel and LaTeX

To write a summation in Word or Excel it is sufficient:

– click on Insert in the multifunctional bar (the first bar at the top);

– click on Equation or Symbol and choose the summation symbol.

Instead, the LaTeX code that allows you to insert a summation is:

\ sum _ {} ^ {} \ left [\ right]

where \ sum inserts the summation symbol, between the first two pairs of curly brackets you enter the values ​​between which the summation index must vary and between \ left [and right], which produce a pair of square brackets , you write l algebraic expression of the summation.

by Abdullah Sam
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