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 ↓


Infinite Series


An infinite series is the sum of infinite terms, which is written as:

∑ a n = a 1 + a 2 + a 3 + ...

where an represents n term in the series. Infinite series are important in mathematical analysis and are used to represent many mathematical phenomena such as functions, calculus, and others.

To begin with, let's consider simple arithmetic operations extended to infinite terms. An infinite series can be convergent or divergent. A convergent infinite series approaches a finite sum as the number of terms tends toward infinity. In contrast, a divergent series does not approach any finite limit as the number of terms grows indefinitely.

Understanding infinite sum with an example

Consider the series:

$ sum_{n=1}^{infty} frac{1}{2^n} = frac{1}{2} + frac{1}{4} + frac{1}{8} + frac{1}{16} + ldots $

It is a geometric progression where the first term a = frac{1}{2} and the common ratio r = frac{1}{2}. The sum S of an infinite geometric progression where |r| < 1 is given by:

$ S = frac{a}{1 - r} $

For our series:

$ S = frac{frac{1}{2}}{1 - frac{1}{2}} = 1 $

It converges geometrically to 1. Despite the infinite nature of the series the sum approaches 1 as the number of terms increases indefinitely.

Visual representation of the series

11/21/4

In the above figure, each colored portion represents halving of the previous portion, visually demonstrating convergence to 1 as squares continue to be added infinitely.

Types of infinite series

  • Geometric series:

    • A series of the form a + ar + ar^2 + ar^3 + ... It converges if |r| < 1.

  • Harmonic Series:

    • A series in the form 1 + frac{1}{2} + frac{1}{3} + frac{1}{4} + ... The harmonic series is known to diverge.

  • P-Series:

    • A series of the form sum_{n=1}^{infty} frac{1}{n^p}. It is convergent if p > 1 and divergent otherwise.

  • Power series:

    • A series of the form sum_{n=0}^{infty} c_n(xa)^n, where each term contains powers of a variable.

Convergence test

Various tests determine whether an infinite series converges:

  1. Ratio test: considers the range:
    2. L = lim (n→∞) |a n+1 / a n |

    If L < 1, the series is convergent. If L > 1, it is divergent. If L = 1, the test is inconclusive.

  2. Root test: considers the limit:
    3. L = lim (n→∞) n√|a n |

    The criteria of convergence and divergence are similar to those of ratio tests.

  3. Comparison test:

    4. If 0 ≤ a n ≤ b n for all n and sum b n is convergent, then sum a n is also divergent. Conversely, if sum a n is divergent, then sum b n is also divergent.

Examples of convergence and divergence

Example 1: Consider the geometric series sum frac{1}{3^n}.

$ frac{a}{1 - r} = frac{frac{1}{3}}{1 - frac{1}{3}} = frac{1}{2} $

Since |r = frac{1}{3}| < 1, it converges to frac{1}{2}.

Example 2: The harmonic series sum frac{1}{n} diverges, which is proved by integration with p = 1 or the p-series test.

Practical applications of infinite series

Infinite series appear ubiquitously in various areas of mathematics and applied sciences such as:

  • Calculus: Infinite series provide function power expansions, like Taylor and Maclaurin series.
  • Physics: quantum mechanics to model phenomena such as wave functions.
  • Engineering: Fourier Series for Signal Processing.

Closing thoughts

Infinite series are fundamental constructs in mathematics and are important for advanced academic fields and a variety of real-world applications. Understanding convergence criteria is important for using these series to approximate functions, solve equations, and accurately model natural phenomena.


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Infinite Series

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