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5.1 Sequences

3 min read•june 24, 2024

Sequences are the building blocks of calculus, forming patterns that stretch to infinity. They can be arithmetic, geometric, or defined recursively, each with its own unique formula and behavior.

Analyzing sequences involves finding limits, determining or , and exploring properties like and boundedness. These tools help us understand how sequences behave as they approach infinity, revealing hidden patterns in mathematical structures.

Sequences

General term formula derivation

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  • Identify the pattern in the given
    • has constant difference between consecutive terms ()
      • an=a1+(n−1)da_n = a_1 + (n - 1)d where a1a_1 is first and dd is common difference
    • has constant ratio between consecutive terms ()
      • General term formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} where a1a_1 is first term and rr is common ratio
  • Determine the values of the constants in the general term formula
    • Use given terms of sequence to set up equations
    • Solve equations to find values of constants (a1a_1, dd, or rr)
    • Plug constants into general term formula to get specific formula for given sequence
  • : defined by relating each term to previous terms
    • Example: Fibonacci sequence Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} with initial conditions F1=F2=1F_1 = F_2 = 1

Sequence limit evaluation

  • Definition of limit lim⁡n→∞an=L\lim_{n \to \infty} a_n = L if for every ε>0\varepsilon > 0, there exists an N∈NN \in \mathbb{N} such that ∣an−L∣<ε|a_n - L| < \varepsilon for all n≥Nn \geq N
  • Determine the by
    • Substituting n→∞n \to \infty into general term formula, if available
      • Simplify resulting expression to find limit value
    • Observing behavior of sequence as nn increases
      • Look for patterns or trends in terms as nn gets large
  • Limits of common sequences
    • lim⁡n→∞c=c\lim_{n \to \infty} c = c where cc is constant (5, -2, π)
    • Sequence approaching finite value lim⁡n→∞an=L\lim_{n \to \infty} a_n = L where LL is finite number (1/2, √3, -7)
    • Sequence approaching infinity lim⁡n→∞an=∞\lim_{n \to \infty} a_n = \infty (n, 2^n, n^2)
    • Sequence approaching negative infinity lim⁡n→∞an=−∞\lim_{n \to \infty} a_n = -\infty (-n, -3^n, -n^3)
  • Determine the existence of a limit
    • If limit exists and is finite value or ±∞\pm \infty, sequence has limit
    • If sequence oscillates (alternates between values) or behaves erratically (no discernible pattern), limit does not exist

Convergence vs divergence analysis

  • Definition of convergence: Sequence {an}\{a_n\} converges if lim⁡n→∞an\lim_{n \to \infty} a_n exists and is finite value
    • Converging sequences approach specific number as nn increases (1/n → 0, (1+1/n)^n → e)
  • Definition of divergence: Sequence {an}\{a_n\} diverges if lim⁡n→∞an\lim_{n \to \infty} a_n does not exist or is ±∞\pm \infty
    • Diverging sequences do not approach specific number (n → ∞, (-1)^n, sin(n))
  • Techniques for determining convergence or divergence
    • : If lim⁡n→∞anbn=L\lim_{n \to \infty} \frac{a_n}{b_n} = L where LL is finite positive number, then {an}\{a_n\} and {bn}\{b_n\} either both converge or both diverge
      • Compare given sequence to known converging/diverging sequence
    • : For sequence {an}\{a_n\}, if lim⁡n→∞∣an+1an∣=L\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L then
      1. If L<1L < 1, sequence converges
      2. If L>1L > 1, sequence diverges
      3. If L=1L = 1, test is inconclusive
    • : For sequence {an}\{a_n\}, if lim⁡n→∞∣an∣n=L\lim_{n \to \infty} \sqrt[n]{|a_n|} = L then
      1. If L<1L < 1, sequence converges
      2. If L>1L > 1, sequence diverges
      3. If L=1L = 1, test is inconclusive

Additional Sequence Properties

  • Monotonicity: describes whether a sequence is increasing, decreasing, or neither
    • Increasing: an≤an+1a_n \leq a_{n+1} for all nn
    • Decreasing: an≥an+1a_n \geq a_{n+1} for all nn
  • : have an upper and lower bound
    • A sequence {an}\{a_n\} is bounded if there exist real numbers MM and mm such that m≤an≤Mm \leq a_n \leq M for all nn
  • : formed by selecting terms from the original sequence while maintaining their order
    • Every subsequence of a converges to the same limit as the original sequence
  • : sequences where terms get arbitrarily close to each other as nn increases
    • A sequence {an}\{a_n\} is Cauchy if for every ε>0\varepsilon > 0, there exists an N∈NN \in \mathbb{N} such that ∣am−an∣<ε|a_m - a_n| < \varepsilon for all m,n≥Nm, n \geq N
    • All convergent sequences are Cauchy sequences

Key Terms to Review (38)

Arithmetic Sequence: An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, and it allows the terms of the sequence to be generated in a predictable pattern.
Bounded above: A sequence is said to be bounded above if there exists a real number M such that every term in the sequence is less than or equal to M. The smallest such M is called the least upper bound or supremum.
Bounded below: A sequence is bounded below if there exists a real number that is less than or equal to every term in the sequence. This means that the terms of the sequence never fall below this specific value.
Bounded sequence: A bounded sequence is a sequence of numbers where all elements lie within a fixed finite interval. Mathematically, a sequence $\{a_n\}$ is bounded if there exists real numbers $M$ and $m$ such that $m \leq a_n \leq M$ for all $n$.
Bounded Sequences: A bounded sequence is a sequence where the values of the terms are confined within a certain range or limit. In other words, the sequence has an upper and lower bound, meaning the terms of the sequence are always less than or equal to a certain positive value and greater than or equal to a certain negative value.
Cauchy Sequences: A Cauchy sequence is a sequence of numbers that converges to a specific limit, where the terms of the sequence become arbitrarily close to one another as the sequence progresses. This concept is fundamental in the study of sequences and series, as it provides a way to determine the convergence of a sequence.
Common Difference: The common difference is a term used in the context of sequences, which are ordered lists of numbers or objects that follow a specific pattern. The common difference refers to the constant value by which each term in the sequence differs from the previous term.
Common Ratio: The common ratio is a constant ratio between consecutive terms in a geometric sequence. It represents the multiplicative factor that is used to generate each successive term from the previous one.
Constant Sequence: A constant sequence is a sequence in which every term is the same value. In other words, a constant sequence is a sequence where the difference between any two consecutive terms is zero, and the ratio between any two consecutive terms is one.
Convergence: Convergence is a fundamental concept in mathematics that describes the behavior of sequences, series, and functions as they approach a specific value or limit. It is a crucial idea that underpins many areas of calculus, including the definite integral, improper integrals, direction fields, numerical methods, sequences, infinite series, and power series.
Convergent sequence: A convergent sequence is a sequence whose terms approach a specific finite value as the index goes to infinity. The value that the terms approach is called the limit of the sequence.
Divergence: Divergence is a fundamental concept in mathematics that describes the behavior of a sequence, series, or function as it approaches or departs from a specific value or pattern. This term is particularly relevant in the context of improper integrals, sequences, infinite series, comparison tests, ratio and root tests, and power series and functions.
Divergence of a series: The divergence of a series occurs when the sum of its terms does not approach a finite limit as more terms are added. A divergent series either increases without bound, decreases without bound, or oscillates indefinitely.
Explicit formulas: An explicit formula directly defines the $n$-th term of a sequence as a function of $n$. It allows for the calculation of any term without needing previous terms.
Fibonacci numbers: Fibonacci numbers form a sequence where each number is the sum of the two preceding ones, starting from 0 and 1. The general formula for the nth Fibonacci number is $F_n = F_{n-1} + F_{n-2}$.
General Term Formula: The general term formula, also known as the explicit formula, is a mathematical expression that describes the nth term or element of a sequence. It provides a concise way to represent the pattern or rule that generates the sequence, allowing for the calculation of any specific term without the need to list out the entire sequence.
Geometric sequence: A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant called the common ratio. The general form of a geometric sequence is $a, ar, ar^2, ar^3,$ and so on.
Geometric Sequence: A geometric sequence is a sequence where each term is a constant multiple of the previous term. The terms in a geometric sequence follow a common ratio, which is the constant multiplier between consecutive terms.
Golden ratio: The golden ratio, often denoted by $\phi$ (phi), is an irrational number approximately equal to 1.618033988749895. It is defined algebraically as $\phi = \frac{1 + \sqrt{5}}{2}$.
Index variable: An index variable is a variable that represents the position of an element in a sequence or series. It is typically denoted by symbols such as $n$, $i$, or $k$.
Infinite sequence: An infinite sequence is a list of numbers in a specific order that continues indefinitely. Each number in the list is called a term.
Limit comparison test: The limit comparison test is a method to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. It involves taking the limit of the ratio of terms from two different series.
Limit Comparison Test: The limit comparison test is a method used to determine the convergence or divergence of a series by comparing it to another series with known convergence or divergence properties. It is a powerful tool for analyzing the behavior of infinite series and sequences.
Limit of a Sequence: The limit of a sequence is the value that the terms of the sequence approach as the index of the sequence increases without bound. It represents the final or stable value that the sequence converges to, provided that such a value exists.
Limit of the sequence: The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. It is denoted as $\lim_{{n \to \infty}} a_n = L$.
Monotone sequence: A monotone sequence is a sequence of numbers that is either entirely non-increasing or non-decreasing. In other words, the terms either consistently increase or consistently decrease.
Monotonicity: Monotonicity is a property that describes the behavior of a function or sequence, indicating whether the values are consistently increasing, decreasing, or staying the same as the independent variable changes. This concept is particularly important in the study of sequences and the application of the Divergence and Integral Tests.
Ratio test: The Ratio Test is used to determine the convergence or divergence of an infinite series by examining the limit of the ratio of successive terms. It is particularly useful for series with factorials or exponential functions.
Ratio Test: The ratio test is a method used to determine the convergence or divergence of a series by examining the behavior of the ratio of consecutive terms. It is a powerful tool for analyzing the convergence of infinite series, particularly power series and sequences.
Recurrence relation: A recurrence relation is an equation that recursively defines a sequence where each term is a function of its preceding terms. It provides a way to generate the terms of the sequence based on previous terms.
Recursive Sequences: A recursive sequence is a mathematical sequence where each term is defined in terms of the preceding terms. This means that the value of a term in the sequence is determined by applying a specific rule or formula to the previous term(s) in the sequence.
Root test: The root test is a method used to determine the convergence or divergence of an infinite series by examining the nth root of the absolute value of its terms. It provides a useful criterion especially when dealing with series where ratio tests are inconclusive.
Root Test: The root test is a method used to determine the convergence or divergence of a series by analyzing the behavior of its terms. It is a powerful tool for studying the properties of power series and evaluating the convergence of sequences.
Sequence: A sequence is an ordered list of elements, typically numbers, that follow a specific pattern or rule. Sequences are fundamental concepts in mathematics, with applications in various fields, including calculus, computer science, and physics.
Subsequences: A subsequence is a sequence that can be obtained from another sequence by deleting some elements without changing the order of the remaining elements. It is a fundamental concept in the study of sequences and their properties.
Term: A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term.
Unbounded sequence: An unbounded sequence is a sequence that does not converge to any finite limit, meaning its terms grow without bound as they approach infinity. In mathematical terms, for any real number $M$, there exists an index $N$ such that for all $n \geq N$, the absolute value of the sequence's terms is greater than $M$.
ε-N Definition: The ε-N definition, also known as the Cauchy sequence definition, is a fundamental concept in the study of sequences in calculus. It provides a precise mathematical way to define the convergence of a sequence, ensuring that the sequence approaches a specific limit as the index increases without bound.
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5.2 Infinite Series