Series in Math: Definition, Example & Application

In mathematics, a series is a sum of the terms of a sequence. Series help us understand patterns and sums in algebra. The concept of series is applied in many areas of mathematics, including algebra, structures, calculus, forming functions, etc. It is significant for kids to have an in-depth understanding of this topic, as it is foundational for many advanced topics like calculus and mathematical analysis. Along with its mathematical applications, the series is widely used in different quantitative disciplines such as physics, computer science, statistics, finance, etc. Let’s learn more about the mathematical series in detail below.

Table of Contents

• What is a series?
• Representation of Series
• Expanded Form
• Sigma Notation
• Types of Series in Mathematics
• Finite Series
• Infinite Series
• Sample Problems Based on Series
• Frequently Asked Questions on Series

What is a Series?

A mathematical series is defined as the sum of the items/terms of a sequence. While a sequence simply lists numbers in a specific order, a series combines those numbers using an addition symbol. Understanding the concept of sequence is important for learning about series. Since a sequence is defined as a list of items arranged in a particular order, a series is the sum of those items. Let’s understand with an example. For a sequence S = 11, 13, 15, 17 of odd numbers, the series would be represented as A:  11 + 13 + 15 + 17 = 56

Representation of Series

Representing a series clearly and accurately is essential for understanding and solving problems in algebra, calculus, and other branches of mathematics. We can represent a series in a few ways, including expanded form and summation notation (sigma notation).

Expanded Form

It is the simplest form of representing a series in which the sum of all terms of a sequence being added is written. 

For a sequence A = a₁, a₂, a₃, …, aₙ, the expression a₁, a₂, a₃, …, aₙ, represents the series for the given sequence.

 

Sigma Notation

We can use a compact form to represent series by using the Greek letter sigma, ∑, to indicate the summation involved.

Thus, the series a₁ + a₂ + a₃, … + aₙ, is abbreviated as ∑nkAk

Types of Series in Mathematics

Series is a foundational concept in various math topics like algebra, calculus, and mathematical analysis, and they help describe patterns, solve equations, and even model real-world problems in science and finance. A series can be finite or infinite, based on whether the given sequence is finite or infinite. 

1. Finite Series 

A series that has only a limited number of terms is called a finite series. Such as 1 + 2 + 3 + 4 = 10

2. Infinite Series 

An infinite series is a series that continues without ending. An example of a finite series is 1 + 3 + 5 + 7 + 9 + ...

 Some of the infinite series converge to approach a sum, while others simply diverge and grow without bound.

3. Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. An arithmetic sequence is the list of terms with a common difference (d).

Formula of arithmetic series: Sₙ = n/2 × (2a + (n - 1)d)

4. Geometric Series

A geometric series multiplies terms by a common ratio (r). 

Formula (finite):

Sₙ = a × (1 - rⁿ) / (1 - r), if r ≠ 1

Solved Problems on Series 

Example 1:

Find the sum of the first 5 terms:

3 + 6 + 9 + 12 + 15

Using the formula:

S₅ = 5/2 × (2×3 + 4×3) = 2.5 × (6 + 12) = 2.5 × 18 = 45 

Answer: 45

 

Example 2:

Sum of first 4 terms: 2 + 4 + 8 + 16

Here, a = 2, r = 2, n = 4

 S₄ = 2 × (1 - 2⁴)/(1 - 2) = 2 × (1 - 16)/(-1) = 2 × (-15)/(-1) = 30

 Answer: 30

 

Frequently Asked Questions  

1: What is the difference between a sequence and a series?

A: A list of numbers; a series is the sum of those numbers.

 

2: What is an infinite series?

A: A series that has an infinite number of terms and may converge or diverge is called an infinite series.

 

3: What is a real-life example of a series?

A: Monthly loan repayments or compound interest calculations are real-life applications of a geometric series.

 

4: What is the symbol used for a series?

A: The Greek letter sigma (∑) is used to represent a sum or series.