Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Real Analysis ↓


Sequence and Series


Introduction

In the study of real analysis, sequences and series are fundamental concepts that form the framework for understanding functions, limits, and continuity. These concepts help us dive into infinity, providing a structured way to handle various mathematical phenomena such as calculus, number theory, and other fields.

Scenes

A sequence is an ordered list of numbers, usually defined by a rule or formula. Mathematically, we denote a sequence as (a_n), where n is a natural number starting at 1 (or sometimes 0). This n represents the position in the sequence. Each number in the list is called a "term".

a_1, a_2, a_3, ..., a_n, ...

Example: Let's consider a simple sequence, where each term is defined by the formula a_n = 1/n.

1, 1/2, 1/3, 1/4, ...

Types of sequences

  • Arithmetic sequence: A sequence in which the difference between consecutive terms is constant. The general term formula is
    a_n = a_1 + (n-1)d
    Example: 2, 4, 6, 8, ... (here, d = 2).
  • Geometric sequence: A sequence in which each term after the first term is found by multiplying the previous term by a fixed, non-zero number called the "common ratio." The common term is the formula
    a_n = a_1 * r^(n-1)
    Example: 3, 6, 12, 24, ... (here, r = 2).
  • Convergent sequence: A sequence that approaches a specific value as n becomes very large. Example:
    1, 1/2, 1/3, 1/4, ...
    This sequence converges to 0.
  • Divergent sequence: A sequence that does not converge to a limit. Example:
    1, 2, 3, 4, ...

Visual representation of a sequence - convergence

Imagine plotting the sequence a_n = 1/n on a graph where the x-axis is the term number n and the y-axis is the value of the term a_n.

One N

As n increases, the points get closer to the x-axis, visually indicating convergence to 0.

Series

A series is the sum of the terms of a sequence. If (a_n) is a sequence, then the series S_n is given by

S_n = a_1 + a_2 + a_3 + ... + a_n

If a sequence of partial sums (S_n) converges to a limit S, then the series is said to be convergent, and S is the sum of the series. Otherwise, it is divergent.

Series can be either finite, where they only include a finite number of terms, or infinite, which continue indefinitely. Infinite series are particularly important in the field of advanced mathematics.

Types of series

  • Arithmetic series: The sum of the terms in an arithmetic sequence. Example:
    S_n = 2 + 4 + 6 + ...
  • Geometric series: The sum of the terms in a geometric sequence. If r is the common ratio, then the series can be expressed as
    S = a_1 / (1 - r) (if |r| < 1)
    Example:
    S = 3 + 6 + 12 + 24 + ...
  • Harmonic series: Typical series where each term is inversely proportional to its position in the sequence:
    S_n = 1 + 1/2 + 1/3 + 1/4 + ...
    This series is divergent, that is, it increases without limit.
  • Power series: Series of this form:
    S = a_0 + a_1*x + a_2*x^2 + ...
    These are often used to represent functions in calculus.

Convergence test

To determine whether an infinite series converges, several tests can be applied:

  • Ratio test: if
    lim (n -> ∞) |a_(n+1)/a_n| = L
    And if L < 1, then the series converges. If L > 1, then it diverges.
  • Basic test: if
    lim (n -> ∞) (|a_n|)^(1/n) = L
    And if L < 1, then the series converges. If L > 1, then it diverges.
  • Integration test: Compare the series to the improper integral ∫f(x)dx. If the integral converges, then the series also converges.
  • Comparison test: Compare with a known benchmark series. If the series a_n is less than the convergent series b_n, then a_n converges.

Visual representation of a series - convergent geometric series

Consider a geometric series in which a_1 = 1 and r = 1/2. Each subsequent term gets progressively smaller. Visually, the series forms a time sequence that decreases exponentially toward the total sum.

,

This sum obviously converges to a finite value, which in this case is S = 2.

Closing thoughts

Sequences and series are crucial components in real analysis and many applications in mathematics, physics, computer science, and engineering. Understanding these ideas allows us to handle infinite processes in a rigorous, structured way. Exploring these concepts is crucial to the understanding of limits, continuity, and differential calculus. Importantly, they bridge the gap between discrete and continuous mathematics, providing a coherent understanding of infinite operations.


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Sequence and Series

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