An ordered collection of objects, called terms or elements, in which repetitions are allowed.

How does a sequence differ from a set?

A sequence preserves order and may include repeated occurrences of an element; a set ignores order and records each element only once.

What does the index of a term indicate?

Its position or rank in the sequence. Indexing commonly begins at either 0 or 1.

What is the length of a finite sequence?

The number of terms it contains. The empty sequence has length 0.

What are lists and streams?

In computing, finite sequences are often called lists, strings, or words, while infinite sequences are often called streams.

Real analysisLesson 3 of 5 in the course Real numbers and sequences

Sequences: Notation, Properties, Convergence, and Series

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A sequence is an ordered collection of objects. Understanding sequences begins with recognizing that order and repeated elements matter, unlike in a set.

A sequence is a collection of objects arranged in a specified order. Its objects are called elements or terms. For example, $(M,A,R,Y)$ has $M$ first and $Y$ last. It is different from $(A,R,M,Y)$ because the order has changed.

All flashcards

What is a sequence?: An ordered collection of objects, called terms or elements, in which repetitions are allowed.
How does a sequence differ from a set?: A sequence preserves order and may include repeated occurrences of an element; a set ignores order and records each element only once.
What does the index of a term indicate?: Its position or rank in the sequence. Indexing commonly begins at either 0 or 1.
What is the length of a finite sequence?: The number of terms it contains. The empty sequence has length 0.
What are lists and streams?: In computing, finite sequences are often called lists, strings, or words, while infinite sequences are often called streams.

Quiz questions

For a recurrence that uses the previous two terms, what is needed to determine the sequence?Answer: Two initial terms.
Which condition on indices defines a subsequence of $(a_n)$ ?Answer: The selected indices increase strictly.
When does the squeeze theorem guarantee that $(c_n)$ converges to $L$ ?Answer: When it is eventually between two sequences converging to $L$ .
How is the value of an infinite series defined?Answer: As the limit of its sequence of partial sums, if that limit exists.
When does the quotient limit law apply to two convergent sequences?Answer: When the denominator sequence has a nonzero limit.

Based on the Wikipedia article Sequence, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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