Alphabeta MathPower Series and Real-Analytic Functions — Examples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
The geometric series represents 1/(1-x) for |x|<1 and re-expands explicitly about every c with |c|<1
Example
The radius-one series with coefficients 1/(n+1)², 1/(n+1) and 1 realise absolute, conditional and divergent endpoint behaviour
Counterexample
The series ∑_n≥0ι(n!)xⁿ converges only at x=0 and has radius zero
Counterexample
The geometric series converges pointwise but not uniformly on (-1,1)
Example
A rational function with nonvanishing denominator is locally represented by geometric-series expansions
Example
The alternating harmonic series illustrates Abel's boundary-limit theorem without evaluating its sum
Counterexample
Grandi's series is Abel summable to 1/2 but its partial sums do not converge
Counterexample
1-2+3-4+⋯ is Abel summable to 1/4 but is not Cesaro summable
Sources
- Power series, Encyclopedia of Mathematics
- Northwestern Math 320-2 lecture notes
- MIT 18.100C, Lecture 11: Power Series
- Cauchy-Hadamard theorem, Encyclopedia of Mathematics
- Analytic function, Encyclopedia of Mathematics
- Abel theorem, Encyclopedia of Mathematics
- S. Semmes, Rice Math 322 notes
- Abel summability, Encyclopedia of Mathematics
- Cesàro summation, Encyclopedia of Mathematics
- Cesàro summation methods, Encyclopedia of Mathematics