Why Doesn't the Sequence 2,0,2,0,2,0 Converge?

In summary, the conversation discusses the use of the word "clearly" in a book on convergence and how it relates to the definition and theorems on convergence. The participants also discuss the need for formal proofs in exams and the use of the triangle inequality in ε-δ proofs.
  • #1
meemoe_uk
125
0
Hi everyone,
I'm doing a course which contains foundation work on convergence.
I was suprised to see the book I am using uses phrases such as...
" This sequence clearly doesn`t converge "
for sequences such as 2,0,2,0,2,0,2,0...
I was expecting it to say something like " By theorem 4.5, this sequence doesn`t converge "
I wouldn`t feel comfortable writing " This sequence clearly doesn`t converge " if, in an exam, I got a question which said " Prove that 2,0,2,0,2,0 doesn`t converge ".
Can anyone point me to basic theorems on convergence which are used to tackle simple questions like this?
 
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  • #2
I don't (like your book) see any reason to appeal to a "theorem".
When your text says "clearly" what it means is that it follows directly from the definition.

A sequence of numbers {an} converges to a limit, L, if, by going far enough on the sequence all the numbers past that point are arbitrarily close to L. Formally: for any [epsilon]>0, there exist an integer N such that if n> N, |an-L|< [epsilon].
("n> N" is "far enough on the sequence", "|an-L|" measures the distance from an to L and "< [epsilon]" is the "arbitrarily close" part.)

Take [epsilon]= 1/2. Two consecutive terms are 2 and 0 and they can't both be with distance 1/2 of anything.
 
  • #3
Well, if I wanted to decide if 2,0,2,0,2 converged then I wouldn`t need to study a bunch of theorems to convince myself it didn`t, because it is clear to my intuition that it doesn`t. But I can`t just write that in an exam. Since I started this maths degree, there's been loads of questions I've been confronted with where the answers are so blatently obvious that I feel like writing " Because it just bloody is! OK? ", but you can`t write that. You've got to apply the fundamental theorems.

Have you attempted a direct proof in what you've written?
Looks OK, part from the last line.
If there's no theorem to fall back on, then I spose I'd have to construct one myself, maybe with induction method.

I like the way you write "theorem", like you think it's a word I've made up.
 
  • #4
Have you attempted a direct proof in what you've written?

Do an indirect proof.

Suppose both 2 and 0 are within distance 1/2 of L.
IOW |2 - L| < 1/2 and |L - 0| < 1/2
Now apply the triangle inequality:
2 = |2 - 0| = |2 - L + L - 0| < |2 - L| + |L - 0| < 1/2 + 1/2 = 1
So 2 < 1
So the supposition was false, and both 2 and 0 cannot be within distance 1/2 from the same number.

(the triangle inequality is one of your best friends when working with &epsilon;-&delta; proofs)
 
  • #5
Thanks hurkyl
 

1. What is "elementary convergence"?

Elementary convergence refers to the process of determining the behavior of a sequence or series of elementary functions, such as polynomials, exponential functions, and trigonometric functions, as the number of terms or iterations approaches infinity.

2. How is elementary convergence different from general convergence?

The main difference is that elementary convergence specifically focuses on sequences or series of elementary functions, while general convergence encompasses a wider range of functions and mathematical concepts.

3. What are some common examples of elementary convergence?

Some common examples include the convergence of geometric series, the Taylor series expansion of a function, and the convergence of the power series for trigonometric functions.

4. What is the importance of studying elementary convergence?

Studying elementary convergence allows us to better understand the behavior of fundamental mathematical functions and their approximations. This can have practical applications in fields such as engineering, physics, and economics.

5. What are some techniques used to prove elementary convergence?

There are various techniques used, such as the comparison test, the ratio test, the root test, and the integral test. These tests help to determine the convergence or divergence of a sequence or series by comparing it to a known function or series with known convergence properties.

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