Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Real Analysis → Sequences and Series ↓


Convergence of Sequences


In the field of mathematics, especially in real analysis, it is essential to understand the concept of convergence in sequences. A sequence is simply an ordered list of numbers. Since sequences are the fundamental components of calculus, they form the backbone of defining and understanding limits, continuity, and other basic concepts of analysis. Let’s dive deep into the convergence of sequences, explore it through definitions, intuitive explanations, examples, and visualizations.

What is sequence?

A sequence is a set of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can be either finite or infinite. In mathematical notation, a sequence is usually written as (a_n), where a_n represents the nth term of the sequence, and n is a positive integer.

For example, consider the simple sequence: 1, 2, 3, 4, 5, ... This sequence can be described by the rule a_n = n, which means that each term is equal to its position in the sequence.

Defining convergence

A sequence (a_n) is said to converge to the limit L if, as n becomes very large, the a_n terms become arbitrarily close to L If such a number L exists, then it is called the limit of the sequence.

Definition: A sequence (a n ) converges to L, and we write: lim (n → ∞) a n = L if for every real number ε > 0, there exists a positive integer N such that: |a n - L| < ε for all n ≥ N.

In simple words, no matter how small a window we choose around L, eventually all the terms of the sequence a_n will lie within this window.

Understanding through examples

Example 1: Convergent sequence

Consider the sequence defined by a_n = 1/n. As n gets larger, 1/n terms get closer and closer to 0. We claim that (1/n) converges to 0.

Given ε > 0, choose N such that 1/N < ε. Then for all n ≥ N, we have: |1/n - 0| = 1/n < 1/N < ε, showing that: lim (n → ∞) 1/n = 0
y = 1/n

Here, as n increases, all the points (n, 1/n) approach the x-axis, which shows convergence to 0.

Example 2: Diverging sequence

Consider the sequence a_n = n. As n gets larger, the terms move away without being bounded by a single number. Therefore, this sequence does not converge; we say that it diverges.

There is no limit L such that the terms approach it. Therefore:

The sequence (a n = n) diverges.

Properties of convergent sequences

Sequences have some interesting properties when they converge:

Specification of boundaries

A sequence can only converge to a limit. If lim (n → ∞) a_n = L and lim (n → ∞) a_n = M, then L = M.

Limitation

Every convergent sequence is bounded. This means that there exist numbers m and M such that for every n m ≤ a_n ≤ M

Examples of limitation

In our previous example (1/n) converges to 0, it is bounded between 0 and 1. Why? Because 0 ≤ 1/n ≤ 1 for all n ≥ 1.

The idea of convergence

Let us imagine with a more general example, the sequence given by the terms a_n = (-1)^n/n. Here, the terms alternate in sign when they are half in magnitude.

y = (-1)^n/n

Note that both the red (n even) and blue (n odd) points approach the line y = 0, so converge to 0.

Subsequences and convergence

A sequence (b_n) is a subsequence of (a_n) if it is formed by deleting some elements of (a_n) without changing the order of the remaining elements. An important result in analysis is:

Bolzano–Weierstrass theorem: Every bounded sequence has a convergent subsequence.

If a sequence (a_n) converges to L, then any subsequence (b_n) of (a_n) also converges to L

Example of subsequence convergence

Consider the sequence a_n = 1/n. A sub-sequence can be b_n = 1/(2n). This also converges to 0 like the original sequence.

lim (n → ∞) 1/(2n) = 0

Challenges in sequence convergence

One may encounter sequences that are neither obviously convergent nor divergent at first glance. To help deal with this complexity we use the concept of a Cauchy sequence.

Cauchy sequence definition

A sequence (a_n) is a Cauchy sequence if for every ε > 0 there exists a positive integer N such that |a_n - a_m| < ε whenever n, m ≥ N

In simple terms, the terms get arbitrarily close to each other as the sequence progresses. The important thing is that in every Euclidean space (like the real numbers), Cauchy sequences are absolutely convergent.

Cauchy sequence behavior

Summary and conclusion

Sequences form the basis for understanding series, continuity, and differentiability. The concept of convergence – whether a series approaches a particular limit or how it might do so – is important to many different fields, from pure mathematics to the applied sciences.

Convergence is not just about sequences getting shorter; it is about the strong idea that no matter how far the sequence travels in infinity, it remains firmly within visual distance of a single value as it continues its journey.

The principles of sequence convergence guide mathematical discovery, ensuring that systems, functions, and results can be understood deeply and completely. As we further explore topics such as series, integrals, and differential equations, the lessons collected about sequences and their convergence will continually prove their importance.


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Convergence of Sequences

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