Summation Notation: A Detailed Explanation with Examples

Another name for summation notation is sigma notation, which is a powerful mathematical tool used to represent and denote the represent the summation series. To express the sum of a sequence, making it easier to work with and understand complex mathematical operations summation notation provides a concise and standardized way. 

In this article we will discuss the definition, and introduction of summation notation, the way of working with sigma notation, writing way summation notation, components, and daily life uses of summation notation. Moreover, we will explain the concept of summation with the help of detailed examples.

Summation Notation

Summation notation, denoted by the sigma (∑) symbol, succinctly represents the sum of a sequence of terms by iterating through an index variable within specified limits.

Example: ∑ⁿ _i=1 a_i sums the terms a₁, a₂, …, a_n using the index variable “I”.

What does the sigma notation demand?

Sigma notation, also known as summation notation, is a concise mathematical representation that employs the Greek letter sigma (∑) to signify the summation or addition of a series of terms. It involves index variable, upper, and lower limits, allowing compact expression of complex sums in various mathematical contexts.

Writing Sigma (Σ) Notation

Expressing a series using sigma notation involves several key steps:

Determine the Expression: Identify the term or sequence you want to sum. Let’s denote this term as “a_i”​.

Choose the Limits: Decide on the range over which you intend to sum. Choose the lower limit (often the starting index) and the upper limit (usually the ending index). The variable that moves through these limits is typically referred to as the index.

Write the Sigma Symbol: Begin with the sigma symbol Σ —, which is a stylized capital letter “S” — and position it at the start of your notation.

Add the Index and Limits: Directly beneath the sigma symbol, jot down the index variable (typically ‘i’), followed by an equals sign =, and then the lower limit. Directly above the sigma, note the upper limit.

Specify the Expression: To the right of the sigma symbol, record the term “a_i”​​ you wish to sum. The index variable “i” is included as a subscript to illustrate its variation across the selected range.

When assembled, the sigma notation “Σ_i=a^b​(a_i)”​represents the sum of the terms a_a​, a_a+1​, …, a_b​, where “i” ranges from a to b.

Components of Summation Notation:

• Sigma (∑) Symbol: The sigma symbol (∑) is the centerpiece of summation notation. It implies that the next phrase is a term sum.

• Index Variable: A variable, usually represented by a letter (often “i” or “n”), is used as an index to iterate through the terms of the series.

• Lower and Upper Limits: The lower limit (usually indicated as a subscript on the sigma symbol) represents the starting value of the index, while the upper limit (usually indicated as a superscript) represents the ending value.

• Expression: The expression to be summed is provided after the sigma symbol. It represents the terms of the series that are being added together.

Examples of Expanding Summation Notation

Expanding summation notation means expressing the terms of a sum individually, rather than in compact sigma (Σ) form. Here are a few examples to illustrate:

Example 1:

Evaluate ∑⁴_x=0(x² + 3x – 1) =?

Solution:

Given series 

∑⁴_x=0(x² + 3x – 1)                                                   (1)

Step 1: We put the value of x one by one to find the value of summation.

For x=0, then we get.

(0)² + 3(0) – 1

0 + 0 – 1

∑⁴_x=0(x² + 3x – 1) = -1

Step 2: For x = 1, then we get

∑⁴_x=0(x² + 3x – 1) = (1)² + 3(1) – 1

= 1 + 3 – 1

= 3

Step 3: Using x =2, then we get.

∑⁴_x=0(x² + 3x – 1) = (2)² + 3(2) – 1

∑⁴_x=0(x² + 3x – 1) = 4 + 6 – 1

∑⁴_x=0(x² + 3x – 1) = 9

Step 4: For x=3, then we get

∑⁴_x=0(x² + 3x – 1) = (3)² + 3(3) – 1

∑⁴_x=0(x² + 3x – 1) = 9 +9 -1

∑⁴_x=0(x² + 3x – 1) = 17

Step 5: For x=4, then we get.

∑⁴_x=0(x² + 3x – 1) = (4)² + 3(4) – 1

∑⁴_x=0(x² + 3x – 1) = 16+12-1

∑⁴_x=0(x² + 3x – 1) = 27

Step 6: Put all step answers in equation (1)

∑⁴_x=0(x² + 3x – 1) = -1 + 3 + 9 + 17 + 27 

∑⁴_x=0(x² + 3x – 1) = 55

To avoid such a lengthy and time-consuming calculations, you can take assistance from a summation notation calculator.

Example 2:

Evaluate summation notation ∑⁴_n=1 f(n) =?

Solution

∑⁴_n=1 f(n) => f (1) + f (2) +f (3) + f (3)

Now, Evaluate 

⇨ (1+1) +(2+1) + (3+1) + (4+1)

⇨ (2) + (3) + (4) + (5)

⇨ 14

Conclusion 

In this article, we have discussed the definition, and introduction of summation notation, the way of working with sigma notation, writing way summation notation, components, and daily life uses of summation notation.

FAQs of Summation notation