Summation notation also known as sigma notation is a mathematical shorthand used to represent the sum of a series of terms. It was first introduced by the Swiss mathematician Leonhard Euler in the **18th century**, revolutionizing the way mathematicians express and manipulate sums.

Before the advent of summation notation, mathematicians had to write out each term in a series individually, which became increasingly cumbersome for long and complex series. Euler’s innovation provided a concise and elegant way to represent such sums.

Summation notation is not limited to simple arithmetic series but can also be used for geometric series, sequences, and other mathematical constructs. It provides a concise way to describe patterns, identify relationships, and perform calculations involving sums.

The adoption of summation notation by mathematicians and scientists has led to significant advancements in various fields, including calculus, number theory, statistics, and physics. Its concise and flexible nature enables mathematicians to express complex ideas more succinctly and make intricate calculations more manageable.

In this article, we will discuss the basic definition of summation notation, and how to perform summation notation with the help of examples in detail.

## What is summation notation?

- Summation notation is a concise mathematical notation that represents the sum of a series of terms by using the sigma symbol ∑ an index of summation and the individual terms of the series.
**Summation notation**is a shorthand way to express the sum of a sequence of numbers or variables, allowing for efficient and compact representation of complex sums.- Summation notation is a powerful technique in mathematics that enables mathematicians to express patterns, calculate sums, and analyze series more easily and concisely.

### Illustration:

These summation notation examples are shown below.

∑^{m}_{y=1 }(y) = 1 + 2 + 3 + 4 +, …, + m

∑^{m}_{y=1 }(y^{2}) = 1^{2 }+ 2^{2} + 3^{2} + 4^{2} +, …, + m^{2}

[∑^{m}_{y=1 }(y)]^{2} = [1 + 2 + 3 + 4 +, …, + m]^{2}

### Summation notation formula

Summation notation, represented by ∑, is used to denote the sum of a series of terms. The universal form of the sigma notation is:

**⇒ ****∑**^{m}_{i}_{=1}**f(x)**

In this formula, i represents the index of summation 1 is the lower limit of the series m is the upper limit and f(x) represents the individual terms of the series.

The notation implies that the variable i takes on values starting from 1 and incrementing by 1 until it reaches m. For each value of i the corresponding term f(x) is added to the overall sum.

Summation notation provides a compact and elegant way to express and compute sums, allowing for efficient manipulation and analysis of series in various **mathematical contexts**.

## How to perform summation notation?

To perform summation notation, you can get help from online tools like a **sigma calculator** to get the result in seconds. Alternatively, you have to follow a step-by-step process that involves evaluating the individual terms of the series and adding them together.

Here’s a general guide on how to perform summation notation manually.

**Identify the limits: **

Determine the lower limit (often denoted as the starting value of the index) and the upper limit (denoted as the ending value of the index) of the series. These limits define the range over which the summation will be performed.

**Set up the notation: **

Write the summation notation with the appropriate index and the terms to be summed. For example, **∑**^{m}_{i=1}**f(x) **represents the sum of terms f(x) = 1,2, 3, …, m

**Substitute values: **

Substitute the values of the index variable i into the expression for each term in the series. Start with the lower limit and increment the index by 1 until you reach the upper limit. Evaluate the corresponding terms for each value of the index.

**Add the terms: **

Sum up the individual terms obtained from step three. Begin with the first term and add each subsequent term to the running total. Continue this process until you have added all the terms.

**Simplify the result: **

If possible, simplify the final sum by combining like terms or applying any applicable mathematical properties or identities.

## How to find summation?

In this section, we will evaluate the example of summation notation.

**Example 1:**

Determine ∑^{4}_{y=1}(y^{3} + 2y + 1)

**Solution:**

**Step 1:**

We write the given function

⇒ y^{3} + 2y + 1

**step 2:**

identify the lower limit to the upper limit

⇒ y = 1 to 5 = 1,2,3,4

**step 3:**

we put the limiting value in the given function

**y = 1**

⇒ y^{3} + 2y + 1 = (1)^{3} + 2(1) + 1 = 1 + 2 + 1 = **4**

**y = 2 **

⇒ y^{3} + 2y + 1 = (2)^{3} + 2(2) + 1 = 8 + 4 + 1 = **13**

**y = 3**

⇒ y^{3} + 2y + 1 = (3)^{3} + 2(3) + 1 = 27 + 6 + 1 = **34**

**y = 4**

⇒ y^{3} + 2y + 1 = (4)^{3} + 2(4) + 1 = 64 + 8 + 1 = **73**

**step 4:**

putting the value of given series

⇒ ∑^{4}_{y=1}(y^{3} + 2y + 1) = 4 + 13 + 34 + 73 = **124**

**Step 5:**

⇒ ∑^{4}_{y=1}(y^{3} + 2y + 1) = **124**

**Example 2:**

Evaluate ∑^{3}_{z=1}(2z^{2} + 3z + 6)

**Solution:**

**Step 1:**

We write the given function

⇒ 2z^{2} + 3z + 6

**Step 2:**

We write the lower limit and upper limit

⇒ Z = 1 to 3 = 1,2,3

**Step 3:**

We put the limiting value and simplify that the series

**Z = 1**

⇒ 2z^{2} + 3z + 6 = 2(1)^{2} + 3(1) + 6 = 2 + 3 + 6 = 11

⇒ 2z^{2} + 3z + 6 = **11**

**Z = 2**

⇒ 2z^{2} + 3z + 6 = 2(2)^{2} + 3(2) + 6 = 2 x 4 + 6 + 6

⇒ 2z^{2} + 3z + 6 = 8 + 6 + 6 = 20

⇒ 2z^{2} + 3z + 6 = **20**

**Z = 3**

⇒ 2z^{2} + 3z + 6 = 2(3)^{2} + 3(3) + 6 = 2 x 9 + 9 + 6

⇒ 2z^{2} + 3z + 6 = 18 + 9 + 6 = 33

⇒ 2z^{2} + 3z + 6 = **33**

**Step 4:**

Also, putting the value of given series

⇒ ∑^{3}_{z=1}(2z^{2} + 3z + 6) = 11 + 20 + 33 = **64**

Step 5:

⇒ ∑^{3}_{z=1}(2z^{2} + 3z + 6) = **64**

## Frequently asked question

**Question 1:**

What does the symbol ∑ represent in summation notation?

**Solution:**

The symbol ∑ in summation notation represents the sum of a series of terms. It is used to compactly denote the addition of multiple terms, where the terms are specified by an index that varies within a given range.

**Question 2:**

How do I read and pronounce summation notation?

**Solution:**

Summation notation is read as “the sum of” or “sigma of.” It is commonly pronounced as “sigma” followed by the expression denoting the terms and the range of summation.

**Question 3:**

How do I calculate the sum of a geometric series using summation notation?

**Solution:**

To calculate the sum of a geometric series using summation notation, you can use the formula:

S = a * (1 – r^{n}) / (1 – r)

By evaluating this summation notation or using the formula, you can calculate the sum of the geometric series efficiently.

### Conclusion

In this article, we have discussed the basic definition of summation notation, the formula of summation notation, and how to perform summation notation in detail. And also, with the help of an example, the topic will be explained. Next, complete studying this article anyone can easily define this topic.