Sigma Notation

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What is Sigma?

Sigma is the eighteenth upper case letter of the ancient Greek alphabet. It is represented as (Σ), also known as sigma notation. As a Greek upper case, sigma notation is used to represent the sum of an infinite number of terms.


In General Mathematics, the lowercase letter (), is generally used to represent unknown angles, as well as, it is a prefix used in different situations to represent that a term is referred in some way to countable unions. For example, a sigma algebra is a group of sets closed under a countable union.


Another common example of the sigma (Σ) is that it is used to represent the standard deviation of the population or a probability distribution, where mu or μ represents the mean of the population).


Sigma Definition

Sigma is the 18th letter of the Greek Alphabet. In the Greek numeral system, sigma has a value of 200. In General Mathematics, the upper case letter (Σ) is used as an operator of the summation, whereas the lower case letter () is used to represent unknown angles.


What Does Sigma Symbol Mean?

The sigma symbol (Σ) is used to represent the sum of an infinite number of terms that follow a pattern.


What is Sigma Function?

Let x be any integer such that x > 1.

The sigma function of positive integer x is defined as the sum of the positive divisor of x. This is generally represented using the Greek letter sigma σ(x). That is

σ(x) = \[\sum_{x/n} d\]

Where \[\sum_{x/n} d\] is the sum of all the positive integer divisors of x.


Here, you can find some of the values of the sigma function.


Integer x

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

σ(x)

1

3

4

7

6

12

8

15

13

18

12

28

14

24

24


What is Sigma Notation?

Sigma notation is a convenient method to represent an infinite number of terms. For example, we often look- forward to summing a number of terms where there is some pattern to the number involved. For example,

1 + 3 + 5 + 7 + 9

Or

1 + 4 + 9 + 16 + 25

The first pattern written above is the sum of the first five odd numbers, whereas the second pattern represents the sum of the first five squared numbers. In other words, if take a sequence of numbers, x1, x2, x3 …….xn, then we can represent the sum of these numbers as:

x1 + x2 + x3 +…….+ xn

The easiest way to write this is to let xk, represent the general term of the sequence, and put

\[\sum_{k=1}^{n} x_{k}\]


Here, the sigma symbol (Σ) is the 18th Greek letter corresponding to our letter S, which means to ‘ sum up’. Hence, the above expression represents the sum of all the terms xk, where k refers to the values from 1 to n. In the above expression, n is the upper limit, and 1is the lower limit.


The sum x1 + x2 + x3 +…….+ xn in sigma notation is represented as:

\[\sum_{i=1}^{n} x_{k}\]


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Sigma Notation Formulas

Here are some of important sigma notation formulas that are frequently used:

  • \[\sum_{i=1}^{x} i = \frac{x(x + 1)}{2}\]

  • \[\sum_{i=1}^{x} k = k_{n}\]

  • \[\sum_{i=1}^{x} i^{2} = \frac{x(x + 1)(x + 2)}{6}\]

  • \[\sum_{i=1}^{n} i^{3} = (\frac{x(x + 1)}{6})^{2}\]

Sigma Notation Examples

Here are some sigma notation example:

  • \[\sum_{i=1}^{n} y^{i}\] = This expression means the sum of the values of y starting at y₁ and ends with yₙ.

\[\sum_{i=1}^{n} y^{i} = y_{1} + y_{2} + y_{3} + . . . . + y_{n}\]

  • \[\sum_{i=1}^{8} y^{i}\] = This expression means the sum of the values of y starting at y₁ and ends with y₈

\[\sum_{i=1}^{8} y^{i} = y_{1} + y_{2} + y_{3} + y_{4} + y_{5} + y_{6} + y_{7} + y_{8}\]

  • \[\sum_{i=2}^{9} y^{i}\] = This expression means the sum of the values of y starting at y₂ and ends with y₉.

\[\sum_{i=2}^{8} y^{i} = y_{2} + y_{3} + y_{4} + y_{5} + y_{6} + y_{7} + y_{8} + y_{9}\]

  • \[\sum_{y=i}^{n} y^{2}i\] = This expression means the sum the squared values of y starting at y₁ and ends with yₙ.

  • \[\sum_{y=i}^{n} y^{2} i = y^{2}_{1} + y^{2}_{2} + y^{2}_{3} + . . . + y^{2}_{n}\]

Arithmetic operations can even be performed on variables within the summation. For example,

\[(\sum_{i=1}^{n} y_{1})^{2}\] = This expression means the sum of the values of y starting at y₁ and ends with yₙ, and square the sum.

\[(\sum_{i=1}^{n} y_{1})^{2} = (y_{1} + y_{2} + y_{3} + . . . + y_{n})^{2}\]

Arithmetic operations can even be performed on expressions including more than one variable. For example,

\[\sum_{i=1}^{n} a_{1}b_{1}\] = This expression represents the product of a and b, starting at a and b, and ends with a and b.

\[a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} + . . . + a_{n}b_{n}\]

\[\sum_{i=1}^{n} k n_{k} = ln\] this expression, k is constant, i.e. an element that does not include a variable of summation and sum includes n elements.


How To Write Series in Sigma Notation?

Consider the following finite arithmetic sequence:

3, 6, 9, 12, 15, 18

Now, add the given terms together (taking the  sum): 3 + 6 + 9 + 12 + 15 + 18

This summation sequence is known as series and is represented by Sn, where n denotes a total number of terms been added.

S6 = 3 + 6 + 9 + 12 + 15 + 18

Sn is often as nth partial sum because it represents a certain part or portion of a sequence.  A partial sum normally starts  a₁ and ends with an, adding n terms.

The summation of a given number of terms of a sequence (series) can also be defined in a compact known as summation notation, sigma notation. The Greek Capital letter also is used to represent the sum.

The series 3 + 6 + 9 + 12 + 15 + 18  can be expressed as \[\sum_{n=1}^{6} 3n\]. This expression is read as a sum of 3n, as n represents numbers from 1 to 6. The variable n is known as the index of summation.

To write the term of series given in sigma notation, replace n by consecutive integers from 1 to 6 as shown below:

\[\sum_{n=1}^{6} 3n = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6)\] 

\[= 3 + 6 + 9 + 12 + 15 + 18\]

\[=  63\]

Let us now understand how to write a series in sigma notation with an example:


Evaluate : \[\sum_{x=1}^{5} (x^{2} + 1)\]

Solution:

Replace x in the expression (x² + 1) with 1, 2, 3, 4, and 5.

\[\sum_{x=1}^{5} (x^{2} + 1) = (1^{2} + 1) + (2^{2} + 1) + (3^{2} + 1) + (4^{2} + 1) + (5^{2} + 1)\]

Note: The expression (x2 + 1) is placed in a set of parentheses following the sigma. Without using the parentheses, only,  x2 would be considered as the part of parentheses, with the plus one( + 1) added on, the sigma is completed.


Sigma Notation Solved Examples

1. Evaluate \[\sum_{y=1}^{5} y^{2}\]

Solution:

The expression given in this example is the sum of all the terms from y = 1 to y = 5. So, we consider each value of x, calculate y3 in each case, and add the result obtained.

\[\sum_{y=1}^{5} y^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2}\]

\[= 1 + 4 + 9 + 16 + 25\]

\[=  55\]


2. Evaluate \[\sum_{n=0}^{6} 3^{n}\]

Solution:

The expression given in this example is the sum of 6 terms because we have n = 0 for the first term.

\[\sum_{n=0}^{6} 3^{n} = 3^{0} + 3^{1} + 3^{2} + 3^{3} + 3^{4} + 3^{5} + 3^{6} + 3^{7}\]

\[= 1 + 3 + 9 + 27 + 81 + 405 + 1215 + 3645\]

\[= 5386\]


Did You Know?

Sigma Notation was introduced by Swiss Mathematician and Physicist Leonhard Euler (1707-1783). He was the first person to use sigma notation using the Greek Letter Σ.

FAQ (Frequently Asked Questions)

Q1. What is Summation Notation?

Ans. A simple method of writing infinite numbers of terms in a sequence is known as summation notation or sigma notation. This includes the 18th Greek letter alphabet. While using sigma notation, the variable written below the sigma is known as the index of summation. The lower number represents the lower limit of the index( where the summation begins) whereas the upper number represents the upper limit of summation( Where the summation ends).

Q2. What is the Sigma Symbol?

Ans. The symbol is used to represent the sum of the number of multiple terms that follow patterns.

For example, the sum of n first whole numbers follows a simple pattern, and can be represented as:

Σk=1n k = 1 +2 + 3 + 4 +....+ n

Q3. What are the Steps to Write a Series in Sigma Notation?

Ans. Following are the steps to write series in Sigma notation:

  • Identify the upper and lower limits of the notation.

  • Substitute each value of x from the lower limit to the upper limit in the formula.

  • Add the terms to find the sum.

For example, the sum of first n terms of a series in sigma notation can be represented as:

Σk=1n xk

This notation asks to find the sum of xk from k=1 to k=n

Here, k is the index of summation, 1 is the lower limit, and n is the upper limit.

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