Real Analysis

[Cthulhu]

Mathematica returns e

[QuantumJG]

Thanks guys.

[Mao]

This question is really bugging me!

Evaluate:



Graphing it on my graphics calculator shows it approaches e if you don't exceed 10¹⁰, but it approaches 1 if you go past 10¹⁴.

WTF!!!!

Yes, it's called truncation error. Your calculator can only store so many digits during calculations, any computations beyond that is unreliable.

[QuantumJG]

Hey guys I'm doing sequence analysis and I just want to have some terms cleared up:

Bounded Sequence:

My understanding of this is:

A sequence is bounded if there exists a

such that:





Increasing or Decreasing Sequence:

My understanding is that:

A sequence is increasing if:

a_n+1 > a_n

A sequence is decreasing if:

a_n+1 < a_n

But what does Cauchy, sup, inf, lim inf, lim sup and Contractive mean?

[QuantumJG]

In a website showing a cauchy example:

this was a part of the example:

|a_m - a_n| < |a_m| + |a_n|

is that from the triangle inequality?

[/0]

We learnt the cauchy sequence as:

Let where is a metric space. Then is a Cauchy sequence if for every , there exists an integer N such that



i.e. as

The terms of a sequence become very 'close' to each other after a certain N.


'sup' is short for 'supremum' or 'least upper bound'. In a partially ordered set ,

If there is an such that and any smaller would not have this property, then

e.g. In a sequence , the supremum would be 1.

The 'infinum' or 'greatest lower bound' is the same thing but from the bottom.

'lim inf' is the least of the limits of the subsequences of

So say you have and there is the limit where . Then the lim inf will be the least such .

Opposite for lim sup.



Also,

is also from the triangle inequality:

Dunno what's contractive

[QuantumJG]

We learnt the cauchy sequence as:

Let where is a metric space. Then is a Cauchy sequence if for every , there exists an integer N such that



i.e. as

The terms of a sequence become very 'close' to each other after a certain N.


'sup' is short for 'supremum' or 'least upper bound'. In a partially ordered set ,

If there is an such that and any smaller would not have this property, then

e.g. In a sequence , the supremum would be 1.

The 'infinum' or 'greatest lower bound' is the same thing but from the bottom.

'lim inf' is the least of the limits of the subsequences of

So say you have and there is the limit where . Then the lim inf will be the least such .

Opposite for lim sup.



Also,

is also from the triangle inequality:

Dunno what's contractive

Cheers.

Ok so I know that Cauchy is when the distance between 2 points as n gets larger and larger, becomes smaller and smaller but how could I prove that something isn't Cauchy?

[/0]

Show that is not necessarily smaller than an epsilon for m,n > N.

e.g.

Then , so for any N you can choose and

Then can be as large as you want, so the sequence can't be Cauchy

[QuantumJG]

Another question,

Am I right by saying that if a sequence is non-Cauchy, then it's divergent?

[Pappa-Bohr]

quantum how are you finding assignment 2?

[QuantumJG]

quantum how are you finding assignment 2?

f$&@ing hard!

How about you?

[/0]

Another question,

Am I right by saying that if a sequence is non-Cauchy, then it's divergent?

Hmm, I think so:

Theorem 8.1.3: A sequence in converges (to a limit in ) iff it is Cauchy.

(the proof is nearly 2 pages)


I'm also battling with my second analysis assignment,,, due tuesday
I spent a few days trying to understand what all the necessary terminology and theorems were... stupid topologist's sine curve...

[Pappa-Bohr]

yea i haven't been able to make much headway (and the terrible structure of our lectures doesn't help at all)

[Pappa-Bohr]

Then

how do you get this? shouldn't there be a 'less than or equal to' sign instead of the equality? (i.e. the triangle inequality)

[QuantumJG]

yea i haven't been able to make much headway (and the terrible structure of our lectures doesn't help at all)

What are you up to?

I'm just starting analysing sequences.