How do you test convergence for an infinite series with imaginary coefficients?

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    Convergence Imaginary
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Discussion Overview

The discussion revolves around methods for testing the convergence of infinite series that include imaginary coefficients. Participants explore different approaches and considerations related to convergence in the context of complex numbers.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests splitting the series into its complex and real parts to test convergence for each separately.
  • Another participant argues that since the reals are a subset of the imaginary numbers, convergence should be checked in the complex plane.
  • A later reply clarifies that the term "imaginary number" typically refers to numbers of the form ri, where r is a real number, and emphasizes that the reals are a subset of the complex numbers.
  • Another participant reiterates the approach of splitting the series into real and imaginary parts, indicating that this leads to dealing with two real-valued series.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between real and imaginary numbers in the context of convergence, and there is no consensus on the best approach to testing convergence for series with imaginary coefficients.

Contextual Notes

Some assumptions about the definitions of real and imaginary numbers may not be explicitly stated, and the implications of these definitions on convergence testing remain unresolved.

jbowers9
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If you have an infinite series some or all of whose coefficients involve imaginary terms how do you check for convergence?
 
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jsut split series into complex and real parts and test convergence for each part
 
The reals are a subset of the imaginary numbers. Hence just check for convergence in the complex plane.
 
ZioX said:
The reals are a subset of the imaginary numbers. Hence just check for convergence in the complex plane.

I wouldn't say that, because when you say imaginary number there is usually the implication that you are speaking of a number in the form ri where r is a real number. The reals are, however, a subset of the complex numbers.
 
like davidsmith said, split up the series into real and imaginary parts. Then you effectively have to deal with two real-valued series.
 

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