Podcast
Questions and Answers
What is the primary difference between a sequence and a series?
What is the primary difference between a sequence and a series?
What is the formula for the sum of an arithmetic series?
What is the formula for the sum of an arithmetic series?
What is the definition of a harmonic sequence?
What is the definition of a harmonic sequence?
What is the condition for a geometric series to be convergent?
What is the condition for a geometric series to be convergent?
Signup and view all the answers
What is the formula for the sum of an infinite geometric series?
What is the formula for the sum of an infinite geometric series?
Signup and view all the answers
What type of sequence is the series 1 + 1/2 + 1/4 + 1/8 + ...?
What type of sequence is the series 1 + 1/2 + 1/4 + 1/8 + ...?
Signup and view all the answers
Study Notes
Sequence and Series
Sequences
- A sequence is an ordered list of numbers, denoted by {a_n} where n is a positive integer.
- Each term in the sequence is denoted by a_n, where n is the term number.
- A sequence can be finite (has a last term) or infinite (has no last term).
Types of Sequences
- Arithmetic sequence: a sequence in which each term is obtained by adding a fixed constant to the previous term.
- Example: 2, 5, 8, 11, ...
- Geometric sequence: a sequence in which each term is obtained by multiplying the previous term by a fixed constant.
- Example: 2, 6, 18, 34, ...
- Harmonic sequence: a sequence in which each term is the reciprocal of an arithmetic sequence.
- Example: 1, 1/2, 1/3, 1/4, ...
Series
- A series is the sum of the terms of a sequence.
- The sum of a series can be finite or infinite.
- A series can be represented as:
- Σa_n = a_1 + a_2 + ... + a_n
Types of Series
- Arithmetic series: the sum of an arithmetic sequence.
- Example: 2 + 5 + 8 + 11 + ...
- Geometric series: the sum of a geometric sequence.
- Example: 2 + 6 + 18 + 34 + ...
- Convergent series: a series that has a finite sum.
- Example: 1 + 1/2 + 1/4 + 1/8 + ...
- Divergent series: a series that has an infinite sum.
- Example: 1 + 2 + 4 + 8 + ...
Formulas
- Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n)
- Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r)
- Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1
Sequences
- A sequence is an ordered list of numbers, denoted by {a_n}, where n is a positive integer.
- Each term in the sequence is denoted by a_n, where n is the term number.
- Sequences can be finite or infinite.
Types of Sequences
- Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
- Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.
- Harmonic sequence: each term is the reciprocal of an arithmetic sequence.
Series
- A series is the sum of the terms of a sequence.
- The sum of a series can be finite or infinite.
- A series can be represented as: Σa_n = a_1 + a_2 +...+ a_n.
Types of Series
- Arithmetic series: the sum of an arithmetic sequence.
- Geometric series: the sum of a geometric sequence.
- Convergent series: a series that has a finite sum.
- Divergent series: a series that has an infinite sum.
Formulas
- Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n).
- Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r).
- Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the basics of sequences and series, including types of sequences such as arithmetic and geometric sequences. Test your understanding of these mathematical concepts!
