Construction of a Cauchy sequence

In summary: Another way could be construct it as follows.Consider $r_{n}$ the area of the regular $n$-gon inscribed into the unit circle.Then the limit is $\pi$.Another way could be construct it as follows.Consider $r_{n}$ the area of the regular $n$-gon inscribed into the unit circle.Then the limit is $\pi$.It is required that every $r_{n}$ must be rational ... for a circle of radious r the area of a regular n sides polygon inscribed is... $\displaystyle A_{n} = \frac{n}{2}\ r^{2}\ \sin \frac{
  • #1
Paradox 101
2
0
I need to construct a cauchy sequence ${r}_{n}$ such that its rational for all n belonging to set of natural numbers, but the limit of the sequence is not rational when n tends to infinity. I know that all convergent sequences are cauchy but I can't randomly conjure up a sequence that satisfies this question. Should I construct the sequence by dedekind cuts? If so how do I go about it? My proof writing isn't good so any help would be appreciated thanks.
 
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  • #2
Welcome to the forum.

As $r_n$ you could take the first $n$ decimal digits of any irrational number.
 
  • #3
Evgeny.Makarov said:
Welcome to the forum.

As $r_n$ you could take the first $n$ decimal digits of any irrational number.

Can you explain this further?
 
  • #4
How about the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... etc.. all the individual terms are rational, having a finite decimal expansion.. but what does it converge to?
 
  • #5
Paradox 101 said:
I need to construct a cauchy sequence ${r}_{n}$ such that its rational for all n belonging to set of natural numbers, but the limit of the sequence is not rational when n tends to infinity. I know that all convergent sequences are cauchy but I can't randomly conjure up a sequence that satisfies this question. Should I construct the sequence by dedekind cuts? If so how do I go about it? My proof writing isn't good so any help would be appreciated thanks.

A good example is the definition of the base of natural logarithm...

$\displaystyle e = \lim_{n \rightarrow \infty} (1 + \frac{1}{n})^{n}\ (1)$

... in which is $\displaystyle r_{n} = (1 + \frac{1}{n})^{n}$...

Demonstrating that e is irrational [not an impossible task...], You obtail a type of sequence You are searching for...

Kind regards

$\chi$ $\sigma$
 
  • #6
Another way could be construct it as follows.

Consider $r_{n}$ the area of the regular $n$-gon inscribed into the unit circle.

Then the limit is $\pi$.
 
  • #7
Fallen Angel said:
Another way could be construct it as follows.

Consider $r_{n}$ the area of the regular $n$-gon inscribed into the unit circle.

Then the limit is $\pi$.

It is required that every $r_{n}$ must be rational ... for a circle of radious r the area of a regular n sides polygon inscribed is...

$\displaystyle A_{n} = \frac{n}{2}\ r^{2}\ \sin \frac{2\ \pi}{n}\ (1)$

If r is rational, then for n=3 is $\displaystyle \sin \frac{2\ \pi}{3} = \frac{\sqrt{3}}{2}$ and $A_{3}$ is irrational... for n=4 is $\displaystyle \sin \frac{\pi}{2} = 1$ and $A_{4}$ is rational...

... regarding other rational sequences for which $\displaystyle \lim_{n \rightarrow \infty} r_{n} = \pi$ it is necessary to consider that a correct definition of an infinite sequence $r_{n}$ consists in defining a procedure that allows for any value of n the computation of $r_{n}$...

Kind regards

$\chi$ $\sigma$
 

Related to Construction of a Cauchy sequence

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any small distance, there is a point in the sequence after which all the terms are within that distance from each other.

2. How is a Cauchy sequence constructed?

A Cauchy sequence is constructed by choosing a starting point (usually denoted by a0), and then defining the terms of the sequence recursively. Each term an is defined in terms of the previous term an-1, such that the difference between consecutive terms becomes smaller and smaller as n increases.

3. What is the importance of Cauchy sequences in mathematics?

Cauchy sequences are important because they help us understand the concept of convergence in mathematics. They are used in the construction of real numbers and are essential in the study of limits and continuity in calculus.

4. How do you prove that a sequence is Cauchy?

To prove that a sequence is Cauchy, one must show that for any small distance, there is a point in the sequence after which all the terms are within that distance from each other. This is typically done using the definition of a Cauchy sequence and the properties of numbers and sequences.

5. Can a sequence be both bounded and Cauchy?

Yes, a sequence can be both bounded and Cauchy. In fact, all bounded sequences are Cauchy sequences. This is because a bounded sequence is one where the terms do not exceed a certain value, and a Cauchy sequence is one where the terms become arbitrarily close to each other as the sequence progresses.

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