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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
- Multi-indexed power series in ℂᵐ and their absolute convergence Definition
- Gelfand transform of ell one of Z Example
- The complex geometric power series has radius 1 and sums to 1/(1-z) for |z|<1 Example
- The geometric series has only one singular point on its unit circle Example
- The power series of z₀/(1-z₁) and the shape of its domain of convergence Example
- A composition of convergent complex power series has a convergent local power-series expansion when the inner sum maps the centre to the outer centre Lemma
- The binomial double series for re-expanding a complex power series is absolutely convergent and may be regrouped Lemma
- The Cauchy kernel expands as an absolutely and uniformly convergent multi-indexed geometric series Lemma
- 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
- A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc Theorem
- An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise Theorem
- 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
- Gleason Kahane Zelazko Theorem
- The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem
Dependency tree · two levels
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Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)