Sigma Notation

Sigma notation is used to write a very long sum (of elements) in a very concise manner. It is NOT used for writing any sum, rather it is the more convenient way of writing the sum of elements that follow a pattern. Sigma notation is referred to as a function as it does a job of adding terms.

Let us learn more about sigma notation along with how to write it using sigma symbol. Let us see the useful formulas that are related to summation.

Sigma notation is the most easiest way of writing a very large sum of elements of a sequence in a simple manner. We know that a sequence is a collection of terms that follow a pattern and sigma notation is used to represent the sum of such elements. This is also known as summation notation as it represents a sum. The terms that are being added in sigma notation are called "summands" or "addends". For example, if we want to write the sum 2 + 4 + 6 + ... + 50 (i.e., the sum of the first 25 even natural numbers) then we can write this sum easily using the sigma notation as \(\sum_{i=1}^{25}\) 2i. This is read as "sigma/summation of 2i where i goes from 1 to 25". Writing the sum using the summation notation was possible because the numbers 2, 4, 6, ..., 50 were following a pattern (or) they belong to the sequence {2i} \(_{i=1}^{25}\). In this notation,

• ∑ is the symbol that is pronounced as "sigma".
• "i" is called the index of summation
• 1 is the first value of n
• 25 is the last value of n.

When we expand the above sigma notation \(\sum_{i=1}^{25}\) 2i, we get 2(1) + 2(2) + .... + 2(25) = 2 + 4 + ... + 50 and we got the actual sum back. We can always cross check our sigma notation by expanding it. In general, the summation notation used to represent the sum of elements of a sequence {a_i}\(_{i=1}^{n}\) looks as follows:

As we have seen in the last section, the sigma symbol in math is ∑ which is pronounced as "sigma". This is one of the Greek alphabets. This sigma symbol is also known as "capital sigma". The summation notation written using the sigma symbol is also known as a "series" as it denotes a sum.

From the above example, you might already have got an idea about how to write the sigma notation. Here are the steps in detail for writing the sum of terms as summation notation.

• Find the general term of the terms of the sum.
  If the sequence of terms is arithmetic / geometric then we can use the general term formula of the respective sequences. If not, we will find the general formula by investigation and observation.
• Select some alphabet (preferably a lowercase letter) to be the index. The usual letters we chose for the index of summation is "i" or "k".
• Observe the sequence and decide the first value and last value of the index.
• Finally, use the sigma symbol to write the sigma notation.

Note that the sigma notation is used to write a sum only when the terms of the sum follow a pattern. Any random sum, say 2 + 3 + 7 + 25 + 51, cannot be written using the sigma as we cannot find a general term in this case. Here is an example.

Example: Write the sigma notation for 16 + 25 + 36 + 49 + .... + 100.

Solution:

It is clear that the sequence {16, 25, 36, 49, ... , 100} is neither arithmetic nor geometric. So we will just observe the terms. We can clearly see that the sequence can be written as squares, i.e., {4², 5², 6², 7², ...., 10²}. If we take the index to be "i", then it is pretty clear that i goes from 4 to 10. Thus, the summation notation for the given sum is \(\sum_{i=4}^{10}\) i².

Expanding the summation notation is just the opposite process of writing it. Here are the steps for writing the same.

• Substitute every value of the index (starting from the first to last and increasing it by 1 every time) in the general term.
• Place a plus symbol between all such terms obtained from the last step.

Example: Let us consider the same sigma notation that we got in the last section \(\sum_{i=4}^{10}\) i². Let us expand it by substituting the values of "i" to be 4, 5, 6, 7, 8, 9, and 10, one by one in the general term i². Then we get 4² + 5² + 6² + 7² + 8² + 9² + 10² which is same as 16 + 25 + 36 + 49 + 64 + 81 + 100.

Expanding the sigma notation helps us to verify whether we have written the notation correctly.

• If there is a constant that is a multiple in the general term, then we can write it out of the notation. i.e., \(\sum_{i={1}}^{n} k a_{i}=k \sum_{i=1}^{n} a_{i}\), where 'k' is a constant.
• The summation notation can be split along the addition or subtraction. i.e.,
  \(\sum_{i={1}}^{n}\left(a_{i} \pm b_{i}\right)=\sum_{i=1}^{n} a_{i} \pm \sum_{i={1}}^{n} b_{i}\)
• The sigma notation cannot be split along the multiplication or division. i.e.,
  \(\sum_{i={1}}^{n}\left(a_{i} b_{i}\right) \neq \sum_{i=1}^{n} a_{i} \sum_{i={1}}^{n} b_{i}\)
  \(\sum_{i={1}}^{n}\dfrac{a_{i}}{ b_{i}} \neq \dfrac{\sum_{i=1}^{n} a_{i}}{\sum_{i={1}}^{n} b_{i}}\)
• \(\sum_{i={1}}^{n} 1\) = 1 + 1 + 1 + ... + 1 (n times) = n.
• \(\sum_{i={1}}^{n} 0\) = 0 + 0 + ... + 0 (n times) = 0.

Though all summations cannot be simplified using some formulas, there are some formulas derived to simplify some famous summation notations. These formulas are very helpful to find the sum of natural numbers, sum of squares of natural numbers, sum of cubes of natural numbers, sum of even numbers, sum of odd numbers, etc. Here is the list of sigma notation formulas:

☛ Related Topics:

• Summation Calculator
• Infinite Series Calculator
• Sum of Arithmetic Sequence Calculator
• Geometric Sequence Calculator

What is Summation Notation?

The summation notation is useful to write the sum of a few or more terms that follow a specific pattern. This is written using a Greek letter called "sigma" and is written as ∑. For example, the sum 3 + 5 + 7 + ... + 21 can be wriiten using the summation notation as \(\sum_{i={1}}^{10}\) (2i + 1). Here, 2i + 1 is the general term of the arithmetic sequence 3, 5, 7, ..., 21.

What is Sigma Symbol in Math?

Sigma symbol is math is a Greek alphabet called "∑" and is read as "sigma". It looks like capital E in the English alphabet. The Sigma symbol is also known as the summation symbol. It is used to write the sum of elements of a sequence.

How Do You Write Sigma Notation Step by Step?

• Identify the general term (a_i) that represents every term of the given sum.
• Identify the lowest value (k) that the general term can take.
• Identify the highest value (h) that the general term can take.
• Then write the sigma notation as \(\sum_{i=k}^h\) (a_i)

For example, we can write 1 + (1/2) + (1/3) + .... + (1/100) as the sigma notation \(\sum_{i=1}^{100}\) (1/i).

What is Sigma Notation Used For?

Sigma notation is just the other way of writing a series. A series is the sum of elements of a sequence. For example, 1 + 8 + 27 + ... + 1000 can be written using sigma notation as \(\sum_{i={1}}^{10}\) i³, as the elements are the cubes of natural numbers from 1 to 10.

How Do You Find Sigma in Math?

To find the sigma in math, we either expand it and find the sum manually or we apply summation formulas to simplify it. For example, \(\sum_{i=1}^5\) i = 1 + 2 + 3 + 4 + 5 = 15.

What is Sigma Notation Formula?

Though formulas are not available to find any type of sigma notations, there are some popular formulas that are helpful to evaluate the summations:

• \(\sum_{i=1}^{n} i\) = 1 + 2 + 3 + ... + n = \(\dfrac{n(n+1)}{2}\)
• \(\sum_{i=1}^{n} i^{2}\) = 1² + 2² + 3² + ... + n² = \(\dfrac{n(n+1)(2 n+1)}{6}\)
• \(\sum_{i=1}^{n} i^{3}\) = 1³ + 2³ + 3³ + ... + n³ = \(\dfrac{n^{2}(n+1)^{2}}{4}\)
• \(\sum_{i=1}^{n}\) 2 i = 2 + 4 + 6 + ... (n numbers) = n (n + 1)
• \(\sum_{i=1}^{n}\) (2 i +1) = 1 + 3 + 5 + .... (n numbers) = n²

What are the Properties of Summation Notation?

Here are some important properties of sigma notation:

• The sigma symbol can be distributed with respect to addition/subtraction but it cannot be distributed with respect to multiplication/division.
• If any constant is multiple in the general term, then it can be written outside the sigma symbol.

How Do You Evaluate Sigma Notation?

To evaluate the sigma notation, just substitute each value of index (from its minimum value to maximum value) with increment of 1 each time into the general term and add all the resultant terms. For example, \(\sum_{i=1}^{5}\) (i² + 1) = (1² + 1) + (2² + 1) + (3² + 1) + (4² + 1) + (5² + 1).