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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?
- A sequence is finite, while a series is infinite.
- A sequence has a fixed constant, while a series has a variable constant.
- A sequence is arithmetic, while a series is geometric.
- A sequence is the list of numbers, while a series is the sum of those numbers. (correct)
What is the formula for the sum of an arithmetic series?
What is the formula for the sum of an arithmetic series?
- Σa_n = (n/2)(a_1 + a_n) (correct)
- Σa_n = (n/2)(a_1 - a_n)
- Σa_n = a_1 / (1 - r), |r| < 1
- Σa_n = a_1(1 - r^n) / (1 - r)
What is the definition of a harmonic sequence?
What is the definition of a harmonic sequence?
- A sequence in which each term is the reciprocal of an arithmetic sequence. (correct)
- A sequence in which each term is obtained by adding a fixed constant to the previous term.
- A sequence in which each term is the sum of the previous two terms.
- A sequence in which each term is obtained by multiplying the previous term by a fixed constant.
What is the condition for a geometric series to be convergent?
What is the condition for a geometric series to be convergent?
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What is the formula for the sum of an infinite geometric series?
What is the formula for the sum of an infinite geometric series?
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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 + ...?
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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.
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