There are various methods used in mathematics, physics, and other subjects to elaborate and understand multiple things. One method can be a supplement to the other methods. While these methods can have completely different structures. Series and sequence are also two such methods that complement each other and are widely used in mathematics.
Series vs Sequence
The main difference between series and sequence is that the series is the product of elements that are used in sequence while a sequence is known to be a specific format of elements in a certain order. Both series and sequence are widely used in multiple branches of mathematics. Also, there are various types of series and sequences.
The sum of the terms found in Sequences is known to be a series. In mathematics, it has a different meaning; that is when the numbers located in sequence are added, their product or some is known as a series. Engineering students use series to signify sound waves and the flow of current.
The sequence is considered to be an event or action that happens one after another. A sequence is a gradual process of events occurring. In other words, a sequence is the number of things happening that are basically arranged in an order. In mathematics, a sequence is known to be a list of numbers in required order.
Comparison Table Between Series and Sequence
| Parameters Of Comparison | Series | Sequence |
| Meaning | When the elements or terms found in sequence add up, the output or sum is known as a series. | When there is a set of elements that are arranged in a certain order, it is known as a sequence. |
| Order | There is no particular order found in series that must be followed like sequences. It is not important. | There is one condition for sequences that is to follow the given order. The element must be in order. |
| Finite series and sequence | The example of a finite series is 1+2+3+4+5+6. | The example of a finite sequence is 1 2 3 4 5 6. |
| Infinite series and sequence | The example of an infinite series is 1+2+3+4….. | The example of an infinite sequence is 1 2 3 4 5….. |
| Types | Mainly there are two types of series in mathematics or some other subjects that are convergent and divergent series. | The sequence has been divided into types are Arithmetic Sequences, Geometric Sequences, Harmonic Sequences, and Fibonacci Numbers. |
What is Series?
When many quantities are added up one after another infinitely in mathematics, it is described as a series. For a series, there should be a definite relation between terms. Series’ are mostly studied and used in Calculus. One can find the existence of series in multiple subjects such as computer science, mathematics, finance, economics, engineering, etc.
In engineering, it is used to study sound waves. In physics, series describe the working progress of a pendulum or how a bouncing ball comes to rest. The example of a series can be a1 + a2 + a3 + a4 + a5. This series can go infinitely. For better understanding, another example can be given. 1 2 3 4 5 are 5 first positive integers. In this form, they are only integers but if they are written as 1+2+3+4+5 = 15, now it will be denoted as a series.
The first summation that described the infinite series was intrigued by Archimedes, a well-known Greek mathematician. His methods are still being used in Calculus. There is a notion called paradoxical in which it is considered that a finite result of a series can be produced by potentially infinite summation. The use of a series totally depends on the subject and its structure changes according to the need.
What is Sequence?
When a particular order is applied to numbers, events, and actions, they are known to be in sequence. A sequence is a significant term used in arithmetic, a branch of mathematics. In mathematics, it is considered to be a list of numbers in the given order. The numbers that are in sequence in mathematics are denoted as ‘terms’. These terms have their definite positions.
To understand it better an example can be taken: ( 6, 11, 18, 22,) is a random sequence of numbers in ascending order. So in this sequence, each number is known as a term and they have their position. For example, 11 has a second position and 22 has the fourth position in this sequence.
Particularly, there are four types of sequence which are Fibonacci Numbers, Arithmetic Sequences, Geometric Sequences, and Harmonic Sequences. In Fibonacci Numbers, two preceding numbers are added to get the sum and such sequences start mostly with 0 and 1. In Arithmetic Sequences, terms are obtained after subtracting and adding up definite numbers.
A definite number gets added or subtracted from these numbers. Hence, each type of arithmetic sequence has a separate definition. In computing, sequences are known as strings and lists. These sequences can be both finite and infinite. Infinite sequences are known as streams.
Main Differences Between Series and Sequence
- The term progression can never become a series. On the other hand, the term progression can become a sequence without exception.
- A series is mostly related to the subjects like Geometric, Arithmetic, Calculus while a sequence can or can not belong to such subjects.
- A series is not required to follow a given order. On the other hand, a sequence is expected to follow the order.
- The sum of elements of a sequence is known as a series while the element must follow a certain order when they are in sequence.
- Series usually have only two types. On the other hand, a sequence is divided into four different types in mathematics.
Conclusion
There are terms in the physical world that can be described to an ordinary person with only words. The above-mentioned terms, series, and sequence have important roles to play in our lives.
With the help of series and sequence, an expert can predict the outcome of a certain event. Series and sequence help individuals monitor those outcomes. Also, they encourage the decision-making of a person. Series and sequence are used a lot in various researches.
These are also used to give analytical reviews of business to the company. Due to their accuracy, series and sequence are adopted widely as analytical tools.
References
- https://ieeexplore.ieee.org/abstract/document/1057620/
- https://pubsonline.informs.org/doi/abs/10.1287/orsc.1.4.375