Calculus How To

Sequence and Series

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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):

  1. Sequence Definition
  2. Series Definition

See also: Series Expansions.

What is a Sequence?

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

  1. The elements (or terms) are in order.
  2. 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 is:

An = a + (n -1)d


  • an 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 an, d and n.

  • an (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

  1. Arithmetic Progression
  2. Bounded Sequence
  3. Cauchy Sequence
  4. Exponential Sequence: Definition, Formula & Examples
  5. Fibonacci Sequence
  6. Finite Geometric Sequence
  7. Infinite Sequence
  8. Infinite Geometric Sequence
  9. Monotonic Sequence / Series
  10. n-tuple
  11. Orthogonal Polynomials
  12. Quadratic Sequence
  13. Sequence of Partial Sums

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

  1. Alternating Series
  2. Alternating Harmonic Series
  3. Asymptotic Series / Poincaré Expansion: Simple Definition, Example
  4. Binomial Series
  5. Continuous Series & Discrete Series
  6. Divergent Series
  7. Finite Series
  8. Fourier Series
  9. Geometric Series
  10. Harmonic Series
  11. Infinite Series
  12. Laurent Series
  13. Monotonic Sequence / Series
  14. Mercator Series
  15. Oscillating Series
  16. P-Series
  17. Positive Series
  18. Power Series
  19. Stirling Series
  20. Telescoping Series
  21. Trigonometric Series, Polynomial: Definitions

Sequence and Series: Related Articles

  1. Alternating Series Test
  2. Pointwise Convergence
  3. Series Rules
  4. Remainder of a Series: Step by Step Example, How to Find
  5. Series Convergence Tests
  6. Sum of a Convergent Geometric Series


Larson, R. & Edwards, B. (2013). Calculus. Cengage Learning; 10 edition

Stephanie Glen. "Sequence and Series" From Calculus for the rest of us!

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