2.1 Definition and Properties of Sequences

Sequences are like mathematical playlists, each number taking its turn in a specific order. They can be finite or infinite, with terms defined by formulas or patterns. Understanding sequences is crucial for grasping limits and series.

Sequences come in various flavors, from arithmetic to geometric, each with unique properties. We'll explore how to identify, define, and analyze these number patterns, setting the stage for deeper concepts in mathematical analysis.

Sequences and Notation

Definition and Representation

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• A sequence is an ordered list of numbers (a1, a2, a3, ..., an)
  • The subscript denotes the position of the term in the sequence
• The notation {an} represents a sequence
  • n is a natural number (positive integer) indicating the term's position
• Sequence terms can be defined explicitly by a formula or implicitly by a recurrence relation
• Sequences can be finite (specific number of terms) or infinite (continuing indefinitely)

Domain and Range

• The domain of a sequence is the set of natural numbers
• The range is the set of values the sequence terms can take
• Example: For the sequence {2n}, the domain is {1, 2, 3, ...}, and the range is {2, 4, 6, ...}

Types of Sequences

Arithmetic and Geometric Sequences

• Arithmetic sequences have a constant difference (d) between consecutive terms
  • General term: $a_n = a_1 + (n - 1)d$
• Geometric sequences have a constant ratio (r) between consecutive terms
  • General term: $a_n = a_1 \times r^{(n-1)}$
• Example: {2, 5, 8, 11, ...} is arithmetic (d = 3), while {2, 6, 18, 54, ...} is geometric (r = 3)

Special Sequences

• Harmonic sequences are defined by the reciprocals of an arithmetic sequence
  • General term: $a_n = \frac{1}{a + (n - 1)d}$, where a and d are constants
• Fibonacci sequence follows the recurrence relation $F_n = F_{n-1} + F_{n-2}$
  • Initial terms: $F_1 = 1$ and $F_2 = 1$
• Constant sequences have the same value for all terms ($a_n = c$ for all n)
• Alternating sequences have terms that alternate in sign (e.g., $(-1)^n$ or $(-1)^{(n+1)}$)

Sequence Properties

Monotonicity

• Monotonically increasing: each term is greater than or equal to the previous term ($a_n \leq a_{n+1}$ for all n)
• Monotonically decreasing: each term is less than or equal to the previous term ($a_n \geq a_{n+1}$ for all n)
• Strictly increasing: each term is strictly greater than the previous term ($a_n < a_{n+1}$ for all n)
• Strictly decreasing: each term is strictly less than the previous term ($a_n > a_{n+1}$ for all n)

Boundedness

• Bounded above: there exists a real number M such that $a_n \leq M$ for all n
• Bounded below: there exists a real number m such that $a_n \geq m$ for all n
• Bounded: both bounded above and bounded below
• Example: The sequence {1/n} is bounded below by 0 and bounded above by 1

General and nth Terms of Sequences

Defining the General Term

• The general term is a formula or expression defining the nth term in terms of n
• For arithmetic sequences: $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term and d is the common difference
• For geometric sequences: $a_n = a_1 \times r^{(n-1)}$, where $a_1$ is the first term and r is the common ratio
• The nth term is found by substituting the value of n into the general term formula

Recursive Sequences and Piecewise Definitions

• Recursive sequences (Fibonacci) require initial terms and the recurrence relation to determine the nth term
• Some sequences have a general term defined piecewise, with different expressions for different ranges of n
• Example: The sequence {an} defined by $a_n = n$ for $n \leq 5$ and $a_n = a_{n-1} + a_{n-5}$ for $n > 5$ is a recursive sequence with a piecewise definition