Find a sequence convering to sqrt(2)

  • Thread starter kidsmoker
  • Start date
  • Tags
    Sequence
In summary, the conversation discusses finding a sequence with a limit of \sqrt{2} using recurrence relations. The suggested method is to use a sequence given by a_{n+1} = \frac{1}{2+a_{n}} and find the positive root of the limit equation. The conversation also mentions the possibility of proving the convergence of the sequence, but it is not required for this module.
  • #1
kidsmoker
88
0

Homework Statement



Find a sequence whose limit is [tex]\sqrt{2}[/tex].

Homework Equations



The work preceeding this was about using recurrence relations to find sequences with desired limits, so that's the method they want me to use.

The Attempt at a Solution



We can find the limit of the sequence given by

[tex]a_{n+1} = \frac{1}{2+a_{n}}[/tex] by noting that [tex]a_{n+1}[/tex] and [tex]a_{n}[/tex] both have the same limit. So we can write

[tex] l = \frac{1}{2+l} [/tex] and find the positive root of that: [tex]l = -1 + \sqrt{2}[/tex]. This is the limit of the sequence.

I can then just add on one to each term to give a sequence with limit [tex]\sqrt{2}[/tex]. Is this all they want me to do you think? Or is there a way to get write another sequence involving [tex]a_{n+1}[/tex] and [tex]a_{n}[/tex] which gives the desired answer?

Thanks.
 
Physics news on Phys.org
  • #2


There are gajillions of ways to answer this problem; yours seems fine.

I do have one question, though: were you merely asked to find a sequence, or are you also expected to prove that the sequence converges to [itex]\sqrt{2}[/itex]? If the latter, I would like to point out that your work shows
If the sequence converges, then it converges to [itex]\sqrt{2}[/itex].​
However, you never showed that the sequence converges, and therefore you cannot conclude that it converges to [itex]\sqrt{2}[/itex].
 
  • #3


Yeah I see what you mean. If it was Analysis then i'd have to prove it by showing it's bounded and increasing or whatever. But this module doesn't seem to bother with that. Thanks.
 

1. How do you find a sequence converging to sqrt(2)?

To find a sequence converging to sqrt(2), we can use the Babylonian method. This involves taking an initial guess (such as 1) and repeatedly applying the formula: (1/2)(x + 2/x), where x is the previous guess. This will gradually approach the value of sqrt(2) as the guesses get closer and closer.

2. Why is it important to find a sequence converging to sqrt(2)?

Finding a sequence converging to sqrt(2) is important in mathematics because it helps us understand the concept of irrational numbers and how they can be approximated by rational numbers. It also has practical applications in fields such as engineering and physics.

3. What is the limit of a sequence converging to sqrt(2)?

The limit of a sequence converging to sqrt(2) is sqrt(2) itself. This means that as the terms in the sequence get closer and closer to sqrt(2), they will eventually reach the exact value of sqrt(2) as the limit.

4. Can you give an example of a sequence converging to sqrt(2)?

One example of a sequence converging to sqrt(2) is (1, 1.4, 1.41, 1.414, 1.4142, ...). Each term in the sequence is a better approximation of sqrt(2) than the previous one, and as the sequence continues, it will eventually reach the exact value of sqrt(2).

5. Is finding a sequence converging to sqrt(2) a difficult task?

It depends on the method used and the level of mathematical understanding. The Babylonian method is relatively straightforward and can be easily understood by most people with some mathematical knowledge. However, finding a precise sequence that converges to sqrt(2) without using a specific method can be a more challenging task.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
767
  • Precalculus Mathematics Homework Help
Replies
7
Views
1K
Replies
1
Views
656
Replies
2
Views
1K
Replies
6
Views
530
  • Calculus and Beyond Homework Help
Replies
7
Views
582
  • Calculus and Beyond Homework Help
Replies
2
Views
89
  • Precalculus Mathematics Homework Help
Replies
17
Views
728
Replies
13
Views
1K
Back
Top