What Test is Used to Find Convergence in Series?

In summary, the test that allows you to find convergence for series such as (0,1,0,1,0,1...) and makes it converge to 1/2 is called Cesaro Summation. However, it only works for series that satisfy the modern definition of convergence. The example given in the conversation (1,-1,1,-1,...) can be solved using Euler's formula for power series, but it only works when -1<x<1. The harmonic series is not the same as the example given. In math, names are important for clear communication of meaning and structure.
  • #1
Savant13
85
1
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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  • #2
That series diverges. You got something wrong. Or maybe I don't understand what you are asking.
 
  • #3
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.
 
  • #4
The arithmetic average of all limit points.
 
  • #5
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.
 
  • #6
I believe the technique you're looking for is called Cesaro Summation...check it out on Wikipedia.
 
  • #7
Yes, Cesaro summation. That's what I was looking for.
 
  • #8
That's a sequence, not a series.
 
  • #9
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
 

FAQ: What Test is Used to Find Convergence in Series?

What is the definition of convergence of a series?

The convergence of a series is when the terms of the series approach a finite value as the number of terms increases, meaning the sum of the series has a finite limit.

How do you test for the convergence of a series?

There are several tests that can be used to determine the convergence of a series, including the comparison test, ratio test, and integral test. These tests analyze the behavior of the series and its terms to determine if it converges or diverges.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the absolute value of each term converges, while conditional convergence refers to a series where the terms alternate in sign and still converge. Absolute convergence is considered stronger because it guarantees that the series converges, while conditional convergence may still result in a divergent series.

Can a series converge to more than one limit?

No, a series can only converge to one limit. If a series has more than one limit, it is considered to be divergent.

How does the rate of convergence affect the convergence of a series?

The rate of convergence describes how quickly the terms in a series approach the finite limit. A series with a faster rate of convergence is considered to converge more quickly, while a series with a slower rate of convergence may take longer to converge.

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