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
   • Sequence Rules and Arithmetic Sequences
   • Nth Term of an Arithmetic Sequence
   • Other Common Types of Sequences
2. Series Definition
   • Common Types of Series

See also: Series Expansions.

What is a Sequence?

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:

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

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

References

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

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