Types of Sequences and Series: Key Concepts with Practical Examples
Sequences and series come in various types. Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio. Harmonic sequences involve the reciprocals of integers, while Fibonacci sequences add the previous two terms. Series, the sum of sequence terms, follow similar classifications: arithmetic series, geometric series, and harmonic series. Convergence is essential for infinite series, where convergent series reach a limit and divergent series do not, shaping applications in calculus and real-world modeling.
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Here’s a summary of each type of sequence with examples:
- Arithmetic Sequence: Constant difference between terms.
Example: \(2, 5, 8, 11, \ldots\) (difference of \(3\). - Geometric Sequence: Constant ratio between terms.
Example: \(3, 6, 12, 24, \ldots\) ratio of \(2\). - Harmonic Sequence: Terms are reciprocals of positive integers.
Example: \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\) - Fibonacci Sequence: Each term is the sum of the two previous terms.
Example: \(0, 1, 1, 2, 3, 5, \ldots\) - Quadratic Sequence: Second differences between terms are constant.
Example: \(3, 7, 13, 21, \ldots\). First differences: \(4, 6, 8, \ldots\); Second differences: \(2, 2, \ldots\) - Arithmetic Series: Sum of an arithmetic sequence.
Example: \(2 + 5 + 8 + 11 + \ldots\) - Geometric Series: Sum of a geometric sequence.
Example: \(3 + 6 + 12 + 24 + \ldots\) - Harmonic Series: Sum of a harmonic sequence.
Example: \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\)
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