Sequence and Series - Calculus How To

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The terms sequence and series sound very similar, but they are quite different. Basically:

A sequence is a set of ordered numbers, like 1, 2, 3, …,
A series is the sum of a set of numbers, like 1 + 2 + 3….

Sequence and Series: Contents (Click to skip to that section):

Sequence Definition
Series Definition
- Common Types of Series

What is a Sequence?

A sequence is a collection of elements (usually numbers) with two major differences from plain old “sets”:

The elements (or terms) are in order.
The values of the terms can repeat.

The number of elements is called the length of the sequence. The length is potentially infinite, meaning it goes on without end. If the collection of numbers ends then it is called a finite sequence.

Sequence Rules and Arithmetic Sequences

Sequences generally have a rule. The rule is applied to find the value of unknown terms. For example, the sequence {3, 6, 9, 12…} begins at 3 and increases by 3 for every subsequent value. This is an example of an arithmetic sequence with a common difference of 3.

The above rule is helpful in defining values in the immediate vicinity of the preceding terms. A limitation of the rule, however, is that it does not tell us the value of an unknown term. In the example above, we don’t know what the 100th term will be. In other words, we need a rule that can quickly determine the value for any unknown term.

Formula and Example for Nth Term

The formula for the nth term of the arithmetic sequence (also called the general term) is:

A_n = a + (n -1)d

Where:

a_n is the nth term of the sequence,
a is the first term,
d is the common difference.

The nth term is the unknown term that you are trying to calculate. Using the formula above you can quickly calculate any nth term.

Example question: Find the 100th term for {3, 6, 9, 12…}:

Step 1: Determine the values for a_n, d and n.

a_n (the nth term that you are trying to find) = 100,
a is the first term: 3,
d is the common difference (3),

Step 2: Place each value from Step 1 into the formula:

a₁₀₀ = 3 + (100 – 1) 3
= 3 + (99) 3
= 3 + 297

a₁₀₀ = 300

The 100th term is 300.

Types of Sequences

Arithmetic Progression

An arithmetic progression (or arithmetic sequence) is a list of numbers, where every term increases by the same amount—called a common difference. In other words, to go from one term to the next, you just add a number. For example, starting with 1:

{1, 2, 3} … add 1 each time,
{1, 7, 14} … add 7 each time.

You don’t have to start with 1 though; You can start with any number. What’s important is that you add the same constant each time.

Finding Common Difference in Arithmetic Progression

The common difference is how much is added to each term in the sequence. For example, the sequence {2, 4, 6, 8} increases by 2—the common difference. Another example: {1, 25, 49} has a common difference of 24. In notation, the common difference is often written as d. For example:

a, a + d, a + 2d, a + 3d, a + 4d.

Which can also be written as a + (n · d), where n starts at zero.

For example, if a = 1 and d = 2:

1, 1 + 2, 1 + 2(2), 1 + 3(2), 1 + 4(2) = 1, 2, 5, 7, 9.

Example question: What is the common difference for the arithmetic progression {5, 8, 11, 14, 17}?

Step 1: Identify the first term. In this list, that’s 5.

Step 2: Subtract the first term from the second: 8 – 5 = 3.

The common difference is 3.

Finite and Infinite Arithmetic Progression

The progression can be a fixed (finite) amount of numbers or an infinite amount. For example, starting with 2 and using a common difference of 3, you get the arithmetic sequence {2, 5, 8, 11…}. The three dots (…) indicate that the sequence goes on an on until infinity. When you have a fixed number of terms, the sequence is called an n-term arithmetic progression. For example, the sequence {2, 5, 8} is a three term arithmetic sequence.

Arithmetic Progression Examples from Number Theory

Arithmetic progression is heavily used in number theory—especially in the analysis of prime numbers. A rather more complex example of an arithmetic progression from number theory:

{a + mk: k >0} ⊆

Where:

m = fixed integer > 0 &
a = fixed integer ≥ 0
- Where:
  - a + mk = equals a prime number,
  - gcd(a, m) = 1.
⊆ = subset of (or equal to),
k = a natural number,
ℕ = set of natural numbers.

The above example is called Dirichlet’s theorem on primes in arithmetic progressions (Lozano-Robledo, 2019). Related to Dirichlet’s theorem are these two important examples of arithmetic progression, which contain all prime numbers except for 2 and 3 (Caldwell, 2020):

1, 7, 13, 19, 25, 31, 37, …
5, 11, 17, 23, 29, 35, 41, …

Sequence and Series: What is a Series?

A series is what you get when you add up the terms in a sequence; when you add these terms, you’re performing what’s more formally called a summation (which is just a fancy way of saying “add them all up”).

Types of Series

Finite Series

A finite series is a sum of a set amount of terms; A series of numbers (e.g. 1 + 2 + 3) is obtained from a sequence of numbers (e.g. 1, 2, 3); Finite series always have a first term and a last term. Plus, you can always find a solution for the sum of a finite series. For example, you can add up a given series of numbers (like 1 + 2 + 3 + 4) and find the answer (10).

A finite series and infinite series only differ from each other in terms of length. You can think of an infinite series as a “… huge or enormously long series” (Lazerowitz & Ambrose, 2016).

More formally, a finite series has the form (Dragomir & Sofo, 2008):

Where:

Σ means to “sum up” (called sigma notation),
a_i; i = 1, 2, …, n) is a sequence of numbers.

Finite Series Example: Finite Arithmetic Series

As a slightly more complicated example, the sum of the numbers 1 through 1000 is a finite sum: 500500:

This particular series is an example of an arithmetic series, which are defined by a common difference between each term (in this example, the difference is 1).

Finite Series Formulas

Some of the more common you’ll come across (Sathaye, 2020):

Name	Formula
Arithmetic Series
Geometric Series
Telescoping Series

Continuous and Discrete Series

The term “continuous series” is used informally, so there isn’t a single definition. Depending on the author, it might mean:

A series with no breaks,
As a synonym for an infinite series,
Where frequencies are given with the variable value (as class intervals).

You’ll want to read up on the authors intention before deciding on an exact definition.

1. A Continuous Series with No Breaks & Discrete Series

A continuous series can be defined as one with has no break or gap in the series. For example, a line segment from two points a to b is the set of all of points between a and b.

Two endpoints A and B on a line segment, representing a continuous series of points from a to b.

Another example: the class of all points (x, y) in a square (including boundaries), arranged in order of magnitude of the x’s (Huntingdon, 2017).

Using the same logic, a discrete series has definite gaps or breaks in between one point and the next. For example, 1, 3, 5, 7,…19 (Singh, 2008).

When used in this sense, what we’re really talking about here is a string of continuous variables (rather than a “continuous series”).

2. Continuous Series with Class Intervals

In this definition (sometimes found in statistics and economics), data is divided into continuous groups. It’s similar to the definition in #1 above, except that the groups are placed into a frequency distribution table.

3. Synonym for Infinite Series

Another definition is simply as another name for “infinite series“. For example, in their article The continuous series of critical points of the two-matrix model at N → ∞ in the double scaling limit, Balaska et al. clearly means infinity (because of the → ∞) in the title. In their 1974 book, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Keith Oldham and Jerome Spanier mention “…a continuous series of continuously differentiable functions,” which likely falls into the realm of this particular definition.

Asymptotic Series / Poincaré Expansion

An asymptotic series (also called an asymptotic expansion or Poincaré expansion) is a way of investigating the general behavior of a function as the independent variable (e.g. x) becomes very large. The procedure, which truncates (cuts off) a function after a certain number of terms (a method of partial sums), can get very good approximations with relatively few calculations — especially when the calculations are so complex it would overwhelm most computers (Beckenbach, 2013).

One of the simplest ways to get an asymptotic series is to do a change of variable x→ 1/x, then perform a series expansion about zero. Other ways to generate these series includes the Euler–Maclaurin summation formula, Integral transforms (e.g. the Laplace transform and Mellin transform) and repeated integration by parts.

As an example, the following expansion can be used to find an approximation for the Riemann zeta function:

Any function smaller that every term in an asymptotic series is subdominant to the series. This means that there isn’t a unique function associated with any particular asymptotic series, but rather one of these series defines an entire class of asymptotically equivalent functions. Theses different asymptotic sequences are called gauges (Paulsen, 2013; Goldstein, 2020).

Sequence and Series: Convergence & Divergence of Asymptotic Series

“One remarkable fact of applied mathematics is the ubiquitous appearance of divergent series, hypocritically renamed asymptotic expansions….The challenge of explaining what an asymptotic expansion is ranks among the outstanding taboo problems of mathematics. —Gian-Carlo Rota, in Indiscrete Thoughts (1997), p. 222″

If you find the formal definitions of asymptotic theory challenging to wrap your head around, you aren’t alone in your confusion. Several prominent authors use the term “asymptotic series” to (incorrectly) describe divergent series (e.g. Dingle, 2013; Gradshteyn & Ryzhik, 2007). In general, while all divergent series are asymptotic series (Cousteix & Mauss, 2007; Erdélyi, 1987), the converse isn’t necessarily true: Asymptotic expansions may or may not be divergent. Therefore, you shouldn’t worry about whether or not a particular asymptotic series will converge or diverge until after you have performed the expansion (Michon, 2020).

Sequence and Series: Related Articles

Sequence and Series: References

Balaska, S. et al. (1998). The continuous series of critical points of the two-matrix model at N → ∞ in the double scaling limit. Nuclear Physics, Section B, Volume 520, Issue 1, p. 411-432.
Beckenbach, E. (2013). Modern Mathematics for the Engineer: Second Series. Dover Publications.
Caldwell, C. (2020). Arithmetic Sequence. Retrieved August 26, 2020 from: https://primes.utm.edu/glossary/page.php?sort=ArithmeticSequence
Cousteix, J. & Mauss, J. (2017). Asymptotic Analysis and Boundary Layers. Springer.
Dingle, R. (2013). Asymptotic Expansions: Their Derivation and Interpretation. Academic Press.
Dragomir, S. & Sofo, A. (2008). Advances in Inequalities for Series. Nova Science Publishers.
Erdélyi, A. (1987). Asymptotic Expansions. New York: Dover,.
Goldstein, R. (2020). Asymptotic versus Convergent series. Optimal truncation. Retrieved November 22, 2020 from: http://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/L3.pdf
Gradshteyn, S. and Ryzhik, I. (2007). Table of Integrals, Series, and Products. Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition.
Huntingdon, E. (2017). The Continuum and Other Types of Serial Order: Second Edition. Courier Dover Publications.
Lozano-Robledo, A. (2019). Number Theory and Geometry. An Introduction to Arithmetic Geometry. American Mathematical Society.
Larson, R. & Edwards, B. (2013). Calculus. Cengage Learning; 10 editionOldham, K. & Spanier, J. (1974). The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier Science.
Lazerowitz, M. & Ambrose, A. (2016). Necessity and Language. Taylor & Francis.
Maor, E. (1991). To Infinity and Beyond. A Cultural History of the Infinite. Princeton University Press.
Michon, G. (2020). Asymptotic Analysis. Retrieved November 20, 2020 from: http://www.numericana.com/answer/asymptotic.htm#series
Paulsen, W. (2013). Asymptotic Analysis and Perturbation Theory. Chapman and Hall/CRC.
Riley, K. et al. (2006). Mathematical Methods for Physics and Engineering. A Comprehensive Guide.
Rota, G. (2000). Indiscrete Thoughts. Birkhäuser.
Sathaye, A. Finite Series Formulas. Retrieved August 9, 2020 from: http://www.msc.uky.edu/sohum/ma110/text/formulas/node2.html
Singh, S. (2008). Biostatistics And Introductory Calculus. Nirali Prakashan.

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Stephanie Glen. "Sequence and Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/sequence-and-series/

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