What is Summation? Definition, examples, and properties

Summation; which means adding up numbers; has been around for a very long time. Early civilizations found ways to calculate and show the total numbers. The ancient Greeks; like Pythagoras and Euclid; also aided us to comprehend how numbers can be added together in a sequence.

Gottfried Wilhelm Leibniz was a mathematician and philosopher, who created the formal symbols and rules for summation in the 17th century. 

In mathematics; notations play an important role to describe and solve complex mathematical problems; summation is one of them that is used to long sum. It is a fundamental concept in mathematics that contains the addition of a sequence of numbers. It is a way to represent the accumulation or total of a set of values.

It is denoted by the Greek letter sigma (∑) and is widely used in many fields of mathematics; including arithmetic; calculus; and beyond. In this comprehensive article; we’ll delve into the definition; History; applications; and properties of Summation; and its significance in different mathematical contexts and other areas of study.

Definition of Summation

A mathematical operation that involves adding together a sequence of numbers or terms is Summation; also known as a series. It represents the accumulation or total of the values within the given sequence. The symbol used to denote summation is the uppercase Greek letter Sigma (∑); which indicates that the following expression is the sum of terms. copyright©iasexpress.net

Symbol of Summation 

The capital sigma symbol (∑) is used to signify summation. It represents that a series of values is being added together. The expression to the right of the symbol indicates the terms being summed, and the limits below and above the symbol define the range or sequence of values involved in the summation. 

How to apply Sigma notation on different series?

Sigma is a versatile tool that may be applied to many mathematical series. Here are some common types of series and how to write them in Sigma notation:

Arithmetic series:

Arithmetic series is a sequence of numbers with a constant difference between consecutive terms. To express it using sigma notation. 

∑ _{i = 1}ⁿ (a + (i – 1) d).

Geometric Series: 

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by Constant ratio (Common ratio). To express it using sigma notation. 

∑ _{i = 0}ⁿ (a × (rⁱ)).

Where:

• a = The first term.
• r = The common ratio.
• n = The number of terms

Harmonic Series:

This series is a sum of reciprocals of positive integers. We can express in sigma notation like this,

∑ _{i = 1}ⁿ ((1 / i)).

Fibonacci Series:

This series is a sequence of numbers where each term is the sum of the two proceeding ones. This can be expressed in sigma notation like this.  copyright©iasexpress.net

∑ _{i = 1}ⁿ (F_i) (b_i).

How to Expand the Sigma Notation? 

Follow these steps to expand the Sigma notation:

1. Identify the lower limit (m) and upper limit (n) of the series.
2. Write down the expression for the function f(i) being summed.
3. Substitute i with each consecutive integer value between m and n.
4. Add up all the terms to get the final result.

Properties of Sigma Notation:

The Sigma notation possesses several properties including some noteworthy are as follows:

Linearity:  This property of summation notation enables us to divide a single summation into multiple summations. This can be expressed as ∑_{i = m}ⁿ [f(i) + g(i)] = ∑ _{i = m}ⁿ f (i) + ∑ _{i = m}ⁿ g (i).

Constant Multiple:  This property allows us to factor out constants from the series. It can be expressed as: 

∑ _{i = m}ⁿ k × f (i) = k × ∑ _{i = m}ⁿ f (i)

Changing the index:  Sometimes, it is more suitable to change the index variable of summation. Let’s say we want to change i to j. The property can be expressed as follows: ∑ _{i = m}ⁿ f (i) = ∑ _{j = m}ⁿ = f (j).

Formulas of the Summation:

1. The sum of First ‘n’ Natural Numbers:

∑ _{i = 1}ⁿ (i) = n × (n + 1) / 2. copyright©iasexpress.net

2. The sum of Squares of First ‘n’ Natural Numbers: 

∑ _{i = 1}ⁿ (i²) = (n × (n + 1) × (2n + 1)) / 6.

3. A sum of Cubes of First ‘n’ Natural Numbers: 

∑ _{i = 1}ⁿ (i³) = (n × (n + 1) / 2)².

4. The sum of Arithmetic Series: 

∑ _{i = 1}ⁿ (a + (i – 1) × d) = n × (a + (a + (n – 1) × d)) / 2.

5. The sum of Geometric Series: 

∑ _{i = 1}ⁿ a × rⁱ = a × (1 – rⁿ) / (1 – r).

How to solve problems of summation notation?

Problems of summation notation can be solved easily with the help of a summation calculator to get the accurate results in a fraction of second. Follow the below examples to manually solve the problems of summation notation.

Example 1: Arithmetic Series

Consider the arithmetic series with a = 2, d = 3, and n = 5. Using sigma notation, represent the sum.

Solution:

∑ _{i = 1}⁵ (2 + (i – 1) × 3) 

Expanding the notation:

Put i = 1, 2, 3, 4, 5

= (2 + (1 – 1) × 3) + (2 + (2 – 1) × 3) + (2 + (3 – 1) × 3) + (2 + (4 – 1) × 3) + (2 + (5 – 1) × 3) 

= 2 + 5 + 8 + 11 + 14 = 40 

Example 2: Geometric Series

Consider the geometric series with a = 2, r = 3, and n = 4. Using sigma notation, represent the sum.

Solution:

∑ _{i = 1}⁴ (2 × 3ⁱ).

Expanding the notation:

= (2 × 3⁰) + (2 × 3¹) + (2 × 3²) + (2 × 3³) + (2 × 3⁴) 

= 2 + 6 + 18 + 54 + 162 

= 242 

Example 4: Fibonacci Series

Consider the Fibonacci series with n = 6. Using sigma notation, represent the sum.

Solution:

∑ _{i = 1}⁶ (Fib(i)).

Expanding the notation:

= (Fib (1)) + (Fib (2)) + (Fib (3)) + (Fib (4)) + (Fib (5)) + (Fib (6)) 

= 1 + 1 + 2 + 3 + 5 + 8 

= 20

Conclusion:

This article explored the history and significance of summation, denoted by the sigma (∑) symbol in mathematics. It represents adding up sequences of numbers, used in various fields like arithmetic and calculus. Examples illustrated how to apply sigma notation in different series, providing a comprehensive understanding of summation. copyright©iasexpress.net