What Test is Used to Find Convergence in Series?

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    Convergence Series
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Discussion Overview

The discussion revolves around identifying the appropriate test for determining the convergence of a specific series, particularly one represented by the sequence (0,1,0,1,0,1...). Participants explore various interpretations of convergence and the terminology used in mathematics, leading to debates about sequences versus series.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asks for the name of the test that shows the series (0,1,0,1,0,1...) converges to 1/2.
  • Multiple participants assert that the series diverges, questioning the initial claim and expressing uncertainty about the question's intent.
  • Another participant suggests that the series might be interpreted as (1,-1,1,-1,...), referencing Euler's formula for power series and its convergence conditions.
  • Cesaro Summation is proposed as a technique that could apply to the series in question, with some participants affirming this identification.
  • There is a distinction made between sequences and series, with some participants emphasizing the importance of correct terminology in mathematical discussions.
  • One participant mentions that the example given was a harmonic series, while others clarify that it was a sequence, not a series.
  • Disagreement arises over the significance of nomenclature versus the underlying mathematical meaning and structure.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the convergence of the series in question, with multiple competing views and interpretations of the terms involved. The discussion remains unresolved regarding the correct classification and convergence of the series.

Contextual Notes

There are limitations in the clarity of definitions and assumptions regarding the terms "series" and "sequence," which contribute to the confusion in the discussion. The mathematical steps and definitions of convergence are not fully resolved.

Savant13
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What is the name of the test that allows you to find convergence for such series as (0,1,0,1,0,1...)? It makes that series converge to 1/2
 
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That series diverges. You got something wrong. Or maybe I don't understand what you are asking.
 
wildman said:
That series diverges. You got something wrong. Or maybe I don't understand what you are asking.

It depends what you mean by diverges.
 
The arithmetic average of all limit points.
 
Savant13 said:
What is the name of the test that allows you to find convergence for such series as (0,1,0,1,0,1...)? It makes that series converge to 1/2
Maybe you mean the series (1,-1,1,-1,...). In that case Euler used the formula for the power series (that (1,x,x^2...) converges to 1/(1-x)) where x=-1. Then he got 1/(1-(-1)) = 1/2. But with the modern definition of convergence, the formula is only true when -1<x<1.
 
I believe the technique you're looking for is called Cesaro Summation...check it out on Wikipedia.
 
Yes, Cesaro summation. That's what I was looking for.
 
That's a sequence, not a series.
 
It's the sequence of partial sums for the series whose terms are (-1)^n (or something close to it), which is not summable by the normal definition, but has Cesaro sum 1/2.
 
  • #10
I suppose I was only considering the first post or so, so never mind. :)

Series are usually written as summations though.
 
  • #11
You're absolutely right about that summation notation would be vastly more appropriate...you definitely have to read a bit into the first post before it makes sense.
 
  • #12
adriank said:
That's a sequence, not a series.

Same thing
 
  • #13
No, it's not the same thing. Please try to learn enough mathematics that you can at least ask an intelligible question without people trying to guess what you are really asking.
 
  • #14
The example I gave was a harmonic series
 
  • #15
No, it was a sequence.

You have a bunch of numbers, a series is a bunch of numbers with an operator.
 
  • #16
And even if you had written it as a series is still would not be a harmonic series! The harmonic series is 1+ 1/2+ 1/3+ ...+ 1/n+ ...
 
  • #17
HallsofIvy said:
No, it's not the same thing. Please try to learn enough mathematics that you can at least ask an intelligible question without people trying to guess what you are really asking.

Dude, this is trivial nomenclature. It's obvious he knows what he means, and it's obvious everyone else knew what he meant, and were able to answer his question. In math it is not the name that is important but rather the meaning and structure.
 
  • #18
maze said:
In math it is not the name that is important but rather the meaning and structure.
That is true we do use mophisms, but in the communication of math name is very important. If you do not communicate clearly the meaning and structure of the objects you describe will be changed.
What is the sum of [sin(x)]^2 by [cos(x)]^2?
[tanh(x)]^2
 

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