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Every absolutely convergent complex series converges, and rearrangements preserve its sum
Statement
Every absolutely convergent complex series converges, and every rearrangement has the same sum. The conventions and prerequisite facts used below are recorded in Complex series, absolute convergence, complex power series, and radius of convergence, An absolutely convergent series in converges, and every rearrangement converges to the same sum.
Facts & Assumptions
Given: A complex series with convergent modulus series.
Proof
Regard its terms as vectors in ; the Euclidean norm is the complex modulus.
Apply the absolute-convergence and rearrangement theorem for finite-dimensional vector series.
Depends on
Used by
- The complex geometric power series has radius 1 and sums to 1/(1-z) for |z|<1 Example
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma
- The complex exponential series converges absolutely for every complex argument Lemma
- Cauchy-Hadamard for complex power series, including zero and infinite radius Theorem
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 123 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)