Difference Between Sequence and Series

Sequence is an ordered collection of elements that follows a specific pattern, whereas a series is a partial sum of sequences. In this article, we will learn the difference between sequence and series, and some of the common misconceptions around it.

Sequence and series are two fundamental mathematical concepts that revolve around a list of numbers and their patterns. But they are often used interchangeably, which leads to confusion. Sequence is an ordered collection of numbers or terms which follows a specified pattern. In contrast, the series refers to the sum of terms in a sequence. Both sequence and series can be finite or infinite.

In this article, we will learn what makes a sequence, what forms a series, and how they differ.

Table of Content

• Sequence vs Series: Difference Between Sequence and Series
• What is Sequence?
  • Properties of Sequence
• What is Series?
  • Properties of Series
• Misconceptions About Sequence and Series

Difference Between Series and Sequence: Series vs Sequence

What is Sequence?

A sequence in mathematics is an ordered collection of elements that follow a specific pattern, with each element assigned a position index. Sequences can be either finite or infinite. 

Example: 

• 1, 3, 5, 7, 9, …
• 2, 4, 6, 8, 10, ….
• -1, 1, -2, 2, -3, 3, -4, 4, ….
• 5, 8, 11, 14, and 17 
• 2, 4, 8, 16, 32, 64, 128,

Properties of Sequence

• A sequence is composed of elements, referred to as a term.
• Each sequence is unique; i.e., with a single change, the sequence can differ.
• The order of the elements in the sequence is always unique.
• A sequence is bounded if a real number M exists, such that all the terms of the sequence are less than M.
• The sequence is convergent if the limit n-> infinity exists.
• A sequence is said to be monotonic if it increases or decreases.

What is a Series?

A series is the sum of the terms in a sequence. A sequence can be changed into a series by simply changing commas into addition signs.

Example: 

• 1 + 2 + 3 + 4 + 5 + …. + 47 + 48 + 49 + 50
• 1/2 + 1/3 + 1/4 + 1/5 + 1/6
• -1 + 1/2 – 1/3 + 1/4 – 1/5 + 1/6 – …..
• 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …..

Properties of Series

• A series is the summation of the terms of a sequence.
• Series can either be finite or infinite. 
• If the sum of the series exists, it equals the limit.
• A series is convergent if the sum of the partial sequence converges.
• Changing the order in which terms are added does not change the sum of the series.
  • This doesn’t hold for infinite series.

Some Common Misconceptions About Sequence and Series

• All Sequences are Series and Vice Versa: This is one of the most common misconceptions, but this is incorrect. A sequence is a list of numbers arranged in a particular order, whereas a series is a partial sum of sequences.
• Sequence and Series are always Infinite: Series and sequence are both finite and infinite. 

Example of Finite Sequence and Series:

Sequence: List of first 9 Natural Numbers: <1, 2, 3, 4, 5, 6, 7, 8, 9>

Series: Sum of natural numbers

• All Series Convereges: The sum of the Series can either converge to a specific number or diverge.

Example: (-1)^n / n 

• The Sum of Infinite Series is Infinite: There are many Series whose sum is finite.

Example: 1/(2^n)

Conclusion

In this article, we have briefly discussed what sequence and series are, the differences between them, their properties, and some of the common misconceptions around series and sequences.

Hope you will like the article.

Keep Learning!!