Calculus How To

Sequence and Series

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

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

Where:

  • 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

Arithmetic Progression
Bounded Sequence: Definition
Fibonacci Sequence
Finite Geometric Sequence
Infinite Sequence
Infinite Geometric Sequence
Monotonic Sequence / Series
n-tuple: Simple Definition, Examples
Orthogonal Polynomials
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

Alternating Series
Alternating Harmonic Series
Binomial Series
Continuous Series & Discrete Series
Divergent Series
Finite Series
Geometric Series
Harmonic Series
Infinite Arithmetic Series
Infinite Series
Laurent Series
Monotonic Sequence / Series
Oscillating Series
P-Series
Positive Series: Definition, Examples, Convergence
Power Series
Telescoping Series

Sequence and Series: Related Articles

Alternating Series Test
Pointwise Convergence
Series Rules
Remainder of a Series: Step by Step Example, How to Find
Sum of a Convergent Geometric Series
Uniform Convergence

References

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

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