ATAR Notes: Forum
Uni Stuff => Science => Faculties => Mathematics => Topic started by: QuantumJG on March 01, 2010, 08:03:32 pm
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Ok I have hit a rather embarrassing but confusing question.
What is x2?
Actually according to wikipedia,
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Ok is it
?
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Another question asks me to define ex


 \times (n-2) \times ... \times 2 \times 1 = 1 \times 2 \times ... \times (n-2) \times (n-1) \times n )

Is this definition ok for ex?
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well the way you have defined e^x is centered at 0, which is a Maclaurin series, it could be centered around other numbers. You could also add that the radius of convergence is infinity, so the series converges for all x.
Maybe you could also use taylor's inequality and prove that e^x is indeed equal to this specific taylor expansion. Which will prove that your definition is correct.
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We haven't looked at that stuff yet! :P
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Oh, it's fun as stuff, omg you will love it when you get to it !!
it's kinda like epsilon - delta proofs, i love it!!
but if you are not up to it, do u know WHY e^x can be expanded like that? I think that is important to know, maybe you can try prove it :P
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Oh, it's fun as stuff, omg you will love it when you get to it !!
it's kinda like epsilon - delta proofs, i love it!!
but if you are not up to it, do u know WHY e^x can be expanded like that? I think that is important to know, maybe you can try prove it :P
Our professor didn't discuss why, but I remember in physics our professor was saying it's from the Taylor series, is that true? Then again he didn't really explain what a Taylor series is.
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if f has a power series expansion:
then the maclaruin series is just a special case of it: namely, centered at 0. (a = 0)
but that's not the slick part, the coolest part is the relationship between the coefficients of the power series f(x) and the derivatives of f(x), thats the part i love the most!!
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Wait until you get to Laurent series in complex analysis and its relation to residues and contour integrals. That's pretty cool ;)
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Ok is it
?
Isn't it just a matter of definition?
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if f has a power series expansion:
then the maclaruin series is just a special case of it: namely, centered at 0. (a = 0)
but that's not the slick part, the coolest part is the relationship between the coefficients of the power series f(x) and the derivatives of f(x), thats the part i love the most!!
So what this is impliying that if,
 = e^{x} )
 = e^{0} = 1)
So
becomes
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if f has a power series expansion:
then the maclaruin series is just a special case of it: namely, centered at 0. (a = 0)
but that's not the slick part, the coolest part is the relationship between the coefficients of the power series f(x) and the derivatives of f(x), thats the part i love the most!!
So what this is impliying that if,
 = e^{x} )
 = e^{0} = 1)
So
becomes
don't you mean when f(x) is centered around 0? not f(a=0)?
it's not implying anything... just that the expansion is "centered" around 0 so that the power series expansion provides a good approximation around 0 xD
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Prove that:
 )
What would be a good way to do this proof?
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\right))
Right angle triangle, opposite is
, hypotenuse is 
Thus tan of this angle is
.
Thus
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What is the difference between
and
?
Also when you are asked:
write sin(x) as an element of
is that just:
sin(x) = 
Also what is:
&
?
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What is the difference between say:
Find the Taylor expansion of ex at x = 0 & Find the Taylor series for sinx at the point a =
?
Help my professor hasn't exactly explained what the Taylor series is.
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Basically it's a polynomial approximation to a function. The more terms it has the more accurate it is.
http://en.wikipedia.org/wiki/Taylor_expansion
Wikipedia has a nice gif of a taylor expansion centred at 0.
If you centre your series at x = 0 then it will be most accurate at x = 0, and as it departs from that point it will start to diverge. Likewise, if you centre at
, it will be very accurate there, but will diverge as you go out.
If you have a taylor series with an infinite number of terms, it doesn't matter where you centre it, since it will be exactly equal to the function within a reasonable domain. However if you want the infinite term expansion then its best to centre it at x = 0 otherwise you will be expanding an infinite number of brackets of the form ^n)
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But what is the difference of a and x?
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a is where the function is centered at.
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Let x
. Graph the sequence:
.
How do I graph this?
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Find the sum of the series:

Can someone tell me what this is asking?
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Find the sum of the series:

Can someone tell me what this is asking?
first of all can you prove its converging series?
if so can u find the infinite sum?
basically what i'm saying is:
if 
then what value does
converge to? [assuming the series
does converge]
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Ok what if I give an example like this:

Approaches 2.25 as n
is this right?
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yeah, basically if you let

then you are forming a new series which the elements are the partial sums of the original sequence.
if the
sequence is converging then the value it converges to is the sum of your original sequence :)
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Ok what if I give an example like this:

Approaches 2.25 as n
is this right?
I understand how you got it, but did you get 2.25 with a calculator?? I mean how are we expected to do these ugly type of q's without a calculator?
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BTW Quantum have you done question 10 from in 'Taylor series' section, of the 'expressions' sheet?
Did you use a proof by induction?
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Ok what if I give an example like this:

Approaches 2.25 as n
is this right?
I understand how you got it, but did you get 2.25 with a calculator?? I mean how are we expected to do these ugly type of q's without a calculator?
So you are in for some fun with real analysis, at the moment I f****en hate the subject and want a professor that actually teaches you how to solve Taylor series problems or even explain what it is!
Yeah I got that value by calculator.
As for question 10 I used induction but wasn't happy with my proof, but hey meh.
This guy is not teaching us the fundementals and I hate it!
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Ok what if I give an example like this:

Approaches 2.25 as n
is this right?
I understand how you got it, but did you get 2.25 with a calculator?? I mean how are we expected to do these ugly type of q's without a calculator?
So you are in for some fun with real analysis, at the moment I f****en hate the subject and want a professor that actually teaches you how to solve Taylor series problems or even explain what it is!
Yeah I got that value by calculator.
As for question 10 I used induction but wasn't happy with my proof, but hey meh.
This guy is not teaching us the fundementals and I hate it!
You shouldn't be doing this if you don't know what a taylor series is lol, better read up quickly!
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Quantum there's a good primer in Stewart Calculus if you've got it
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yeah the chapter in stewarts on sequence and series is very well set out.
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So you are in for some fun with real analysis, at the moment I f****en hate the subject and want a professor that actually teaches you how to solve Taylor series problems or even explain what it is!
Yeah I got that value by calculator.
As for question 10 I used induction but wasn't happy with my proof, but hey meh.
This guy is not teaching us the fundementals and I hate it!
Yea it's a bit weird how he doesn't teach us the actual concepts...., I mean he just does random questions from the problem sheet, his website 'notes' are just a mess as-well lol..
I've read a bit of (Baby) Rudin and Spivak to help elucidate the concepts, and that's helped a little bit, but I hate self-learning mathematics from textbooks.
The other thing I find really bizarre is the length of this assignment - almost 80 questions long!, when the first assignment from last year only had 8 questions...
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Surely you covered taylor series and series in general last year?
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Surely you covered taylor series and series in general last year?
No it wasn't covered!
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Pappa-Bohr can you back me up that we really haven't covered Taylor series.
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your lecturer is an idiot if you haven't covered taylor series amd he/she expects you to do these questions...
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Ok I was looking through the lecture outline and apparently we go into more detail further down the track.
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your lecturer is an idiot if you haven't covered taylor series amd he/she expects you to do these questions...
We've covered examples of the questions, so we can most them, but we haven't actually covered the theory or purpose behind them.
I mean it's like we're still waiting for the proper theory lectures to start, and up and till now we've been doing warm up questions or something..
It's a very oddly structured subject.
but hey we have a different lecturer for the next few weeks so maybe it will get better.
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So have you been able to do q70?
He hasn't shown us how to find the sum of a series.
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i just did it on calculator and got 0.346574
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Yeah but how can we get that without a calculator?
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i dont know.
post it and see if anyone else can do it?
also did you do the q 60 in the Taylor series section?
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i dont know.
post it and see if anyone else can do it?
also did you do the q 60 in the Taylor series section?
I did question 60 and I'll give you a hint:
Find the series expansion of ex in powers of 'x+3'.
As for question 70, I'm going to just leave it.
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Did you do calculus last year? Taylor series aka power series aka power expansion (etc etc) is one of the first things they teach you in single-variable calculus at uni, along with convergence and divergence of series, it was even covered in enhancement.
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This question is really bugging me!
Evaluate:
^{n} )
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!!!!
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Isn't that another definition for e?
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Mathematica returns e
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Thanks guys.
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This question is really bugging me!
Evaluate:
^{n} )
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.
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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:
an+1 > an
A sequence is decreasing if:
an+1 < an
But what does Cauchy, sup, inf, lim inf, lim sup and Contractive mean?
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In a website showing a cauchy example:
this was a part of the example:
|am - an| < |am| + |an|
is that from the triangle inequality?
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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
<\varepsilon)
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 _{n \geq 1})
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
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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
<\varepsilon)
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 _{n \geq 1})
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?
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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
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Another question,
Am I right by saying that if a sequence is non-Cauchy, then it's divergent?
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quantum how are you finding assignment 2?
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quantum how are you finding assignment 2?
f$&@ing hard!
How about you?
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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 :o
I spent a few days trying to understand what all the necessary terminology and theorems were... stupid topologist's sine curve...
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yea i haven't been able to make much headway (and the terrible structure of our lectures doesn't help at all)
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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)
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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.
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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)
I don't know if you're learning about metric spaces, but in general, a metric
is similar to the notion of 'distance' between x and y.
If we use the standard euclidean metric
. This essentially is the 'distance' between two numbers, and it is what is normally used in sequences.
Thus we wish for a cauchy that for
,
:
For 
Then
(assuming of course
).
And since we can set
,
etc etc we see that it is not cauchy
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Thanks /0. In a tute yesterday I was told what those things meant.
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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)
The proof of that can be made simple to the point that you can recreate it yourself, if you just split it into little pieces and treat it as a sequence of little lemmas, e.g: prove for
first and have
as just a corollary.
I'm gonna make this conjecture now that this thread got me thinking(I've been in the mood of making conjectures lately, so far half are true): Suppose
are complete metric spaces, then the metric space
with metric
is complete.
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wat did u get for sheet 1 section 7 question 15.
i.e ''limit as n goes to infinity'' for 1+(-1)^(n+1)
(also is n assumed to be a positive integer or a real or what?)
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What is the difference between the maxima of a set and the supremum of a set?
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The supremum of a set
is the least possible upper bound of the set. The supremum need not be in the set.
The maximum of a set
is a number
which is an upper bound of the set, i.e.
,
. The maximum must be an element of the set.
As a result, some kinds of sets such as open sets don't have maximums. All finite partially ordered sets have maximums.
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What does the standard topology of a metric space mean?
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The usual toplogy on a metric space is one where the open sets are those sets that are the unions of open balls, along with the empty set.
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Approaches 2.25 as n
is this right?
Not sure if its still relevant, but you should be able to recognise this sum as a relative of the Geometric series - Consider the derivative of
. (Have you learnt about uniform continuity and power series?)
edit: fixed sum index.