Sequence vs. Series: Know the Difference
By Shumaila Saeed || Published on January 31, 2024
A sequence is an ordered list of numbers following a specific rule, while a series is the sum of the elements of a sequence.
Key Differences
Sequence refers to an ordered list of numbers where the order follows a specific pattern or rule. Each number in the sequence is called a term. For example, the sequence of even numbers starts as 2, 4, 6, 8, ... In contrast, a Series is the sum of the terms of a sequence. For instance, the series of the first four even numbers is 2 + 4 + 6 + 8.
Shumaila Saeed
Jan 31, 2024
Sequence can be finite or infinite, depending on whether it has a limited or unlimited number of terms. For example, the sequence of the first five prime numbers (2, 3, 5, 7, 11) is finite. However, a Series can also be finite or infinite, like the sum of the first five prime numbers (2 + 3 + 5 + 7 + 11) or the sum of all natural numbers.
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Jan 31, 2024
The nature of the rule defining a Sequence can vary, including arithmetic, geometric, or based on a more complex formula. An arithmetic sequence increases by a constant, like 3, 6, 9, 12, ... Conversely, in a Series, this pattern determines how the terms are added together, such as the arithmetic series 3 + 6 + 9 + 12 + ...
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Jan 31, 2024
In a Sequence, each individual term has its own identity and position in the order. For example, in the Fibonacci sequence (1, 1, 2, 3, 5, ...), each number is distinct. However, in a Series, the individual identity of the terms is less important than their collective sum. The sum of the first five Fibonacci numbers is 1 + 1 + 2 + 3 + 5.
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Jan 31, 2024
The study of Sequences is fundamental in various fields of mathematics, particularly in calculus and number theory, focusing on the properties of the individual terms and their patterns. In contrast, Series are crucial in areas like analysis and complex calculations, focusing on the behavior of the sum of sequences, especially when the number of terms is very large or infinite.
Shumaila Saeed
Jan 31, 2024
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Comparison Chart
Definition
Ordered list of numbers following a specific rule
Sum of the elements of a sequence
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Jan 31, 2024
Example
2, 4, 6, 8, ... (even numbers)
2 + 4 + 6 + 8 + ... (sum of even numbers)
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Focus
On the individual terms and their order
On the collective sum of the terms
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Application
Used in calculus, number theory
Used in analysis, summing infinite sequences
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Jan 31, 2024
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Sequence and Series Definitions
Sequence
An ordered list of numbers.
The Fibonacci sequence starts with 1, 1, 2, 3, 5, 8.
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Series
Summation of a sequence of numbers.
The series 1 + 2 + 3 + 4 + 5 totals 15.
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Sequence
Collection of terms with identifiable patterns.
The geometric sequence 3, 6, 12, 24, ... doubles each term.
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Series
The cumulative total of a number sequence.
In the series 2 + 4 + 6, each term adds even numbers.
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Sequence
A series of numbers with a specific order.
The sequence of square numbers is 1, 4, 9, 16, ...
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Series
Aggregate sum of terms in a specific order.
The geometric series 1 + 2 + 4 + 8 sums to 15.
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Sequence
A progression of numbers based on a formula.
In the sequence 2n+1, we get 3, 5, 7, 9, ...
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Series
An operation adding numbers in a sequence.
The series formed by adding square numbers 1 + 4 + 9 is 14.
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Series
A number of objects or events arranged or coming one after the other in succession.
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Series
Physics & Chemistry A group of objects related by linearly varying successive differences in form or configuration
A radioactive decay series.
The paraffin alkane series.
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Sequence
(Games) Three or more playing cards in consecutive order and usually the same suit; a run.
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Series
(Mathematics) The sum of a sequentially ordered finite or infinite set of terms.
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Sequence
A series of related shots that constitute a complete unit of action in a movie.
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Series
(Geology) A group of rock formations closely related in time of origin and distinct as a group from other formations.
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Sequence
(Music) A melodic or harmonic pattern successively repeated at different pitches with or without a key change.
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Sequence
Roman Catholic Church A hymn sung between the gradual and the Gospel.
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Series
A succession of publications that present an extended narrative, such as a comic book series, or that have similar subjects or similar formats, such as a series of cookbooks.
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Series
A succession of individual programs presented as parts of a unified whole, such as the set of episodes of a television show or a podcast.
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Sequence
(Biochemistry) The order of constituents in a polymer, especially the order of nucleotides in a nucleic acid or of the amino acids in a protein.
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Series
(Sports) A number of games played by the same two teams, often in succession.
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Sequence
To determine the order of constituents in (a polymer, such as a nucleic acid or protein molecule).
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Series
(Linguistics) A set of vowels or diphthongs related by ablaut, as in sing, sang, sung, and song.
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Series
A number of things that follow on one after the other or are connected one after the other.
A series of seemingly inconsequential events led cumulatively to the fall of the company.
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Sequence
(uncountable) The state of being sequent or following; order of succession.
Complete the listed tasks in sequence.
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Series
(broadcasting) A television or radio program which consists of several episodes that are broadcast in regular intervals
“Friends” was one of the most successful television series in recent years.
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Sequence
A series of musical phrases where a theme or melody is repeated, with some change each time, such as in pitch or length (example: opening of Beethoven's Fifth Symphony).
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Series
(mathematics) The sequence of partial sums of a given sequence ai.
The harmonic series has been much studied.
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Sequence
A musical composition used in some Catholic Masses between the readings. The most famous sequence is the Dies Irae (Day of Wrath) formerly used in funeral services.
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Series
A group of matches between two sides, with the aim being to win more matches than the opposition.
The Blue Jays are playing the Yankees in a four-game series.
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Sequence
(mathematics) An ordered list of objects, typically indexed with natural numbers.
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Series
(botany) A subdivision of a genus, a taxonomic rank below that of section (and subsection) but above that of species.
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Sequence
A series of shots that depict a single action or style in a film, television show etc.
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Sequence
(card games) A meld consisting of three or more cards of successive ranks in the same suit, such as the four, five and six of hearts.
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Series
(phonology) A set of consonants that share a particular phonetic or phonological feature.
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Series
(electronics) Connected one after the other in a circuit, in series.
You have to connect the lights in series for them to work properly.
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Sequence
To determine the order of things, especially of amino acids in a protein, or of bases in a nucleic acid
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Series
A number of things or events standing or succeeding in order, and connected by a like relation; sequence; order; course; a succession of things; as, a continuous series of calamitous events.
During some years his life a series of triumphs.
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Series
Any comprehensive group of animals or plants including several subordinate related groups.
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Sequence
The state of being sequent; succession; order of following; arrangement.
How art thou a kingBut by fair sequence and succession?
Sequence and series of the seasons of the year.
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Series
In Engler's system of plant classification, a group of families showing certain structural or morphological relationships. It corresponds to the cohort of some writers, and to the order of many modern systematists.
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Sequence
That which follows or succeeds as an effect; sequel; consequence; result.
The inevitable sequences of sin and punishment.
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Series
An indefinite number of terms succeeding one another, each of which is derived from one or more of the preceding by a fixed law, called the law of the series; as, an arithmetical series; a geometrical series.
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Sequence
Simple succession, or the coming after in time, without asserting or implying causative energy; as, the reactions of chemical agents may be conceived as merely invariable sequences.
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Series
A mode of arranging the separate parts of a circuit by connecting them successively end to end to form a single path for the current; - opposed to parallel. The parts so arranged are said to be in series.
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Sequence
Any succession of chords (or harmonic phrase) rising or falling by the regular diatonic degrees in the same scale; a succession of similar harmonic steps.
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Sequence
A hymn introduced in the Mass on certain festival days, and recited or sung immediately before the gospel, and after the gradual or introit, whence the name.
Originally the sequence was called a Prose, because its early form was rhythmical prose.
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Series
Similar things placed in order or happening one after another;
They were investigating a series of bank robberies
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Sequence
Three or more cards of the same suit in immediately consecutive order of value; as, ace, king, and queen; or knave, ten, nine, and eight.
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Series
A serialized set of programs;
A comedy series
The Masterworks concert series
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Sequence
The specific order of any linear arrangement of items; as, the sequence of amino acid residues in a protein; the sequence of instructions in a computer program; the sequence of acts in a variety show.
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Sequence
To determine the sequence of; as, to sequence a protein or a DNA fragment.
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Series
(sports) several contests played successively by the same teams;
The visiting team swept the series
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Sequence
Serial arrangement in which things follow in logical order or a recurrent pattern;
The sequence of names was alphabetical
He invented a technique to determine the sequence of base pairs in DNA
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Series
A group of postage stamps having a common theme or a group of coins or currency selected as a group for study or collection;
The Post Office issued a series commemorating famous American entertainers
His coin collection included the complete series of Indian-head pennies
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Sequence
A following of one thing after another in time;
The doctor saw a sequence of patients
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Series
(mathematics) the sum of a finite or infinite sequence of expressions
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Sequence
Film consisting of a succession of related shots that develop a given subject in a movie
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Series
(electronics) connection of components in such a manner that current flows first through one and then through the other;
The voltage divider consisted of a series of fixed resistors
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Series
A sequential addition of numbers.
The series of the first three prime numbers is 2 + 3 + 5.
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Sequence
Determine the order of constituents in;
They sequenced the human genome
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Sequence
A succession of numbers following a rule.
An arithmetic sequence like 5, 10, 15, 20 adds 5 each time.
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Jan 05, 2024
Repeatedly Asked Queries
What is the main difference between a sequence and a series?
A sequence is an ordered list of numbers, while a series is the sum of these numbers.
Shumaila Saeed
Jan 31, 2024
Can a sequence be infinite?
Yes, sequences can be either finite or infinite.
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Jan 31, 2024
What is the use of sequences in mathematics?
Sequences are used to study patterns and solve problems in calculus and number theory.
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Jan 31, 2024
Can a series have negative terms?
Yes, if the sequence it is summing has negative terms.
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Jan 31, 2024
Are all series summable?
Not all; some infinite series may not converge to a finite sum.
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Jan 31, 2024
What is a convergent series?
A series where the sum of the terms approaches a finite limit.
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Jan 31, 2024
What is a harmonic sequence?
A sequence where each term is the reciprocal of an integer.
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Jan 31, 2024
How is a series related to its sequence?
A series is the sum of the terms of its corresponding sequence.
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Jan 31, 2024
What is an arithmetic sequence?
A sequence where each term increases by a constant.
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Jan 31, 2024
What is a geometric series?
A series where each term is multiplied by a constant to get the next term.
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What is an example of a real-life series?
Calculating interest compounded annually is an example of a series.
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Jan 31, 2024
What's a famous example of a sequence?
The Fibonacci sequence is a well-known example.
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Jan 31, 2024
Can sequences be graphed?
Yes, sequences can be represented graphically, with each term plotted on a graph.
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Jan 31, 2024
Can a sequence be non-numeric?
Typically, sequences in mathematics are numeric, but in other contexts, like in computer science, they can be non-numeric.
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Jan 31, 2024
How important is the order in a sequence?
Very important, as it defines the pattern or rule of the sequence.
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Jan 31, 2024
How do you find the sum of a series?
The sum depends on the type of series; for arithmetic and geometric series, there are specific formulas.
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Jan 31, 2024
Can sequences and series predict real-world phenomena?
Yes, they are used in various fields like physics, economics, and biology for modeling.
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Jan 31, 2024
Is there a series for every sequence?
Yes, any sequence can have a corresponding series, but its sum may not always be meaningful or finite.
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Jan 31, 2024
What role do series play in calculus?
Series are crucial in calculus for understanding functions and limits.
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Jan 31, 2024
Is the concept of series used in statistics?
Yes, especially in the calculation of statistical measures and probabilities.
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Jan 31, 2024
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About Author
Written by
Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.
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Hifza Nasir
