MHB Construction of a Cauchy sequence

[Paradox 101]

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.

 

[Evgeny.Makarov]

Welcome to the forum.

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

 

[Paradox 101]

Welcome to the forum.

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

Can you explain this further?

 

[Nono713]

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?

 

[chisigma]

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

$\chi$ $\sigma$

 

[Fallen Angel]

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 $\pi$.

 

[chisigma]

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

$\chi$ $\sigma$