In mathematics, a sequence is a list of numbers that follows a specific pattern or rule. A series is the sum of the terms in a sequence. Both sequences and series play a crucial role in various fields of mathematics, including calculus, number theory, and analysis. Let’s dive deeper into the world of sequences and series.
What Are Sequence and Series?
Sequences
A sequence can be defined as an ordered list of numbers in which each number in the list is called a term. The terms of a sequence can be finite or infinite, and they can be represented using various notations, including a closed-form formula, recursive formula, or using sequence brackets. A sequence is said to converge if its terms approach a finite value as the sequence progresses. If a sequence does not converge, it is said to diverge.
Series
A series is the sum of the terms in a sequence. In other words, a series is the sum of an infinite number of terms in a sequence. There are various types of series, including arithmetic series, geometric series, and harmonic series. Just like sequences, series can also converge or diverge. If a series converges, the sum of the infinite terms is called the limit of the series.
Difference Between Sequence and Series
| Sequence | Series |
|---|---|
| A sequence is an ordered list of numbers in which each number follows a specific pattern or rule. | A series is the sum of the terms in a sequence. |
| A sequence can have a finite or infinite number of terms. | A series always has an infinite number of terms. |
| Each term in a sequence is represented by a specific notation or formula. | A series is represented as the sum of the terms in a sequence. |
| A sequence can converge or diverge. | A series can also converge or diverge. |
| The sum of the terms in a sequence is not calculated. | The sum of the terms in a series is calculated. |
| Examples of sequences include the Fibonacci sequence and the prime number sequence. | Examples of series include the arithmetic series, the geometric series, and the harmonic series. |
Types of Sequence and Series
There are various types of sequences and series, which is given below:
Types of Sequences:
- Arithmetic Sequence
- Geometric Sequence
- Harmonic Sequence
- Fibonacci Sequence
- Triangular Sequence
- Square Number Sequence
- Cubic Number Sequence
- Prime Number Sequence
- Square Root Sequence
- Exponential Sequence
Types of Series:
- Arithmetic Series
- Geometric Series
- Harmonic Series
- Power Series
- Taylor Series
- Fourier Series
- Laurent Series
- Dirichlet Series
- Alternating Series
- Convergent and Divergent Series
Arithmetic Sequence and Series Formula
| Arithmetic sequence | a, a + d, a + 2d, a + 3d, … |
|---|---|
| Arithmetic series | a + (a + d) + (a + 2d) + (a + 3d) + … |
| First term: | a |
| Common difference(d): | Successive term – Preceding term or an – an-1 |
| nth term, an | a + (n-1)d |
| Sum of arithmetic series, Sn | (n/2)(2a + (n-1)d) |
Geometric Sequence and Series Formulas
| Geometric sequence | a, ar, ar2,….,ar(n-1),… |
|---|---|
| Geometric series | a + ar + ar2 + …+ ar(n-1)+ … |
| First term | a |
| Common ratio | r |
| nth term | ar(n-1) |
| Sum of geometric series | Finite series: Sn = a(1−rn)/(1−r) for r≠1, and Sn = an for r = 1
Infinite series: Sn = a/(1−r) for |r| < 1, and not defined for |r| > 1 |
Sequence and Series Examples
Examples of Sequences:
- Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
- Arithmetic Sequence: 2, 5, 8, 11, 14, 17, 20, …
- Geometric Sequence: 2, 4, 8, 16, 32, 64, …
- Harmonic Sequence: 1, 1/2, 1/3, 1/4, 1/5, …
- Prime Number Sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
Examples of Series:
- Arithmetic Series: 5 + 10 + 15 + 20 + 25 = 75
- Geometric Series: 2 + 4 + 8 + 16 + 32 = 62
- Harmonic Series: 1 + 1/2 + 1/3 + 1/4 + 1/5 + … = ∞
- Power Series: 1 + x + x^2 + x^3 + x^4 + … = 1/(1-x) (|x| < 1)
- Alternating Series: 1 – 1/2 + 1/3 – 1/4 + 1/5 – … = ln(2)
FAQs on Sequence and Series
What is a sequence in math?
A sequence is an ordered list of numbers in which each number follows a specific pattern or rule. Each term in a sequence is represented by a specific notation or formula.
What is a series in math?
A series is the sum of the terms in a sequence. It is represented as the sum of the terms in a sequence and can have an infinite number of terms.
What is an arithmetic sequence?
An arithmetic sequence is a sequence in which each term is obtained by adding a constant value to the previous term. The common difference between each term is constant.
What is a geometric sequence?
A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant value. The constant ratio between each term is constant.
What is an arithmetic series?
An arithmetic series is the sum of the terms in an arithmetic sequence. The formula for the sum of the first n terms in an arithmetic series is: S_n = (n/2)(a_1 + a_n), where a_1 is the first term, a_n is the nth term, and n is the number of terms.
What is a geometric series?
A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first n terms in a geometric series is: S_n = (a_1(1 – r^n))/(1 – r), where a_1 is the first term, r is the common ratio, and n is the number of terms.
What are some real-world applications of sequence and series?
Sequence and series have many applications in the real world, including finance, physics, engineering, and computer science. For example, arithmetic and geometric sequences are used to calculate loan payments and interest rates, while power series are used in calculus to represent functions as infinite sums of simpler functions.