Glossary

2.1 Definition and Properties of Sequences

3 min read•july 30, 2024

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: an=a1+(n−1)da_n = a_1 + (n - 1)d
  • Geometric sequences have a constant ratio (r) between consecutive terms
    • General term: an=a1×r(n−1)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
    • General term: an=1a+(n−1)da_n = \frac{1}{a + (n - 1)d}, where a and d are constants
  • follows the recurrence relation Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}
    • Initial terms: F1=1F_1 = 1 and F2=1F_2 = 1
  • Constant sequences have the same value for all terms (an=ca_n = c for all n)
  • Alternating sequences have terms that alternate in sign (e.g., (−1)n(-1)^n or (−1)(n+1)(-1)^{(n+1)})

Sequence Properties

Monotonicity

  • : each term is greater than or equal to the previous term (an≤an+1a_n \leq a_{n+1} for all n)
  • : each term is less than or equal to the previous term (an≥an+1a_n \geq a_{n+1} for all n)
  • : each term is strictly greater than the previous term (an<an+1a_n < a_{n+1} for all n)
  • : each term is strictly less than the previous term (an>an+1a_n > a_{n+1} for all n)

Boundedness

  • Bounded above: there exists a real number M such that an≤Ma_n \leq M for all n
  • Bounded below: there exists a real number m such that an≥ma_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 in terms of n
  • For arithmetic sequences: an=a1+(n−1)da_n = a_1 + (n - 1)d, where a1a_1 is the first term and d is the common difference
  • For geometric sequences: an=a1×r(n−1)a_n = a_1 \times r^{(n-1)}, where a1a_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 an=na_n = n for n≤5n \leq 5 and an=an−1+an−5a_n = a_{n-1} + a_{n-5} for n>5n > 5 is a recursive sequence with a piecewise definition

Key Terms to Review (18)

Absolute convergence: Absolute convergence refers to a series that converges when the absolute values of its terms are summed. This concept is crucial because if a series converges absolutely, it guarantees that the series converges regardless of the arrangement of its terms, linking it to various properties and tests of convergence for series and functions.
Alternating Sequence: An alternating sequence is a sequence of numbers in which the terms alternately take positive and negative values. This characteristic leads to interesting properties regarding convergence and boundedness, particularly in mathematical analysis, where understanding the behavior of sequences is crucial for determining limits and their overall properties.
Arithmetic sequence: An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This common difference can be positive, negative, or zero, and it plays a crucial role in determining the behavior of the sequence as it progresses. Understanding arithmetic sequences helps to analyze their monotonic properties, limits, and foundational aspects of sequences more broadly.
Bolzano-Weierstrass Theorem: The Bolzano-Weierstrass Theorem states that every bounded sequence in $b{R}^n$ has a convergent subsequence whose limit lies within the same space. This theorem is fundamental in understanding the properties of sequences and functions, particularly in the context of continuity and optimization.
Bounded Sequence: A bounded sequence is a sequence of numbers where there exists a real number that serves as an upper limit and another real number that serves as a lower limit, meaning all terms of the sequence fall within this range. This concept is crucial for understanding the behavior of sequences, especially when analyzing their convergence and divergence, as well as their relationships with monotonicity and Cauchy properties.
Cauchy Convergence Criterion: The Cauchy convergence criterion states that a sequence is convergent if and only if, for every positive real number $$\epsilon$$, there exists a natural number $$N$$ such that for all natural numbers $$m, n \geq N$$, the distance between the terms of the sequence is less than $$\epsilon$$. This concept emphasizes that the terms of a convergent sequence become arbitrarily close to each other as the sequence progresses, which is a crucial property when examining sequences.
Convergent Sequence: A convergent sequence is a sequence of numbers that approaches a specific value, called the limit, as the index goes to infinity. This concept connects to the behavior of functions and limits, highlighting how sequences can be analyzed using various limit theorems and properties. Understanding convergent sequences is crucial for grasping the foundational ideas in mathematical analysis, especially in relation to Cauchy sequences and completeness.
Divergent Sequence: A divergent sequence is a sequence of numbers that does not converge to a specific limit as the sequence progresses to infinity. Instead, the terms of the sequence either grow without bound, oscillate, or behave erratically, resulting in no single value to which they settle. Understanding divergent sequences is essential for grasping the broader concepts of limits, especially in the study of sequences and their behaviors.
Fibonacci sequence: The Fibonacci sequence is a mathematical series where each number is the sum of the two preceding ones, typically starting with 0 and 1. This sequence illustrates important properties of sequences, as it highlights patterns of growth and recursion, connecting to broader concepts in analysis such as limits and convergence.
Harmonic Sequence: A harmonic sequence is a sequence of numbers where the reciprocals of the terms form an arithmetic sequence. This means that if you take the reciprocal of each term in the harmonic sequence, those values will have a constant difference. Harmonic sequences can often be found in various mathematical contexts and are used to analyze relationships between terms, particularly in calculus and series.
Limit of a sequence: The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. Understanding this concept is essential for analyzing the behavior of sequences and helps in deriving important results related to convergence, continuity, and differentiability in mathematical analysis.
Limiting Behavior: Limiting behavior refers to the tendency of a sequence to approach a specific value as the index of the sequence increases. This concept is crucial in understanding how sequences behave as they progress toward infinity, allowing for the determination of convergence or divergence, which in turn helps in analyzing various properties of sequences.
Monotonically decreasing: A sequence is said to be monotonically decreasing if each term is less than or equal to the preceding term, meaning that as you progress through the sequence, the values either decrease or stay the same. This property is crucial in understanding the behavior of sequences, as it indicates a consistent trend of reduction. Monotonically decreasing sequences can converge to a limit and are important when analyzing bounded sequences and their convergence characteristics.
Monotonically increasing: A sequence is said to be monotonically increasing if each term in the sequence is greater than or equal to the preceding term. This property indicates that as you move through the sequence, the values do not decrease, which can be crucial for understanding the behavior and limits of sequences. A monotonically increasing sequence may eventually converge or diverge, but it retains a consistent trend of non-decrease.
Nth term: The nth term refers to a general formula that describes the position of an element in a sequence, allowing one to determine any term's value based on its position. This concept is fundamental in understanding how sequences are constructed and analyzed, as it provides a way to express patterns within the terms. By using the nth term, we can simplify the process of finding specific terms without needing to enumerate all previous terms in the sequence.
Strictly decreasing: A sequence is strictly decreasing if each term is less than the previous term. This means that for a sequence {a_n}, it holds that a_n > a_{n+1} for all n. This property implies that as you move through the terms of the sequence, the values become progressively smaller, showcasing a clear trend of decline without any plateaus or increases.
Strictly increasing: A sequence is considered strictly increasing if each term is greater than the preceding term. This means that for a sequence {a_n}, it holds true that if n < m, then a_n < a_m. This characteristic indicates that as you move through the sequence, the values are continuously rising without any repeats or plateaus, which has important implications for convergence and limits.
Subsequence: A subsequence is a sequence derived from another sequence by deleting some or no elements without changing the order of the remaining elements. This concept is crucial because it allows for the analysis of the properties of sequences, particularly in understanding limits and convergence behavior. By focusing on subsequences, one can explore how certain properties hold true or break down, especially in limit theorems.
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