Convergence of Series

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

That series diverges. You got something wrong. Or maybe I don't understand what you are asking.

lurflurf
Homework Helper
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.

daniel_i_l
Gold Member
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.

I suppose I was only considering the first post or so, so never mind. :)

Series are usually written as summations though.

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.

That's a sequence, not a series.
Same thing

HallsofIvy
Homework Helper
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.

The example I gave was a harmonic series

No, it was a sequence.

You have a bunch of numbers, a series is a bunch of numbers with an operator.

HallsofIvy
Homework Helper
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+ ....

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.

lurflurf
Homework Helper
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