Real Analysis

[QuantumJG]

What is the difference between say:

Find the Taylor expansion of e^x 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.

[/0]

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

[QuantumJG]

But what is the difference of a and x?

[TrueTears]

a is where the function is centered at.

[QuantumJG]

Let x . Graph the sequence:

.

How do I graph this?

[QuantumJG]

Find the sum of the series:



Can someone tell me what this is asking?

[TrueTears]

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]

[QuantumJG]

Ok what if I give an example like this:



Approaches 2.25 as n is this right?

[TrueTears]

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

[Pappa-Bohr]

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?

[Pappa-Bohr]

BTW Quantum have you done question 10 from in 'Taylor series' section, of the 'expressions' sheet?
Did you use a proof by induction?

[QuantumJG]

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!

[TrueTears]

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!

[/0]

Quantum there's a good primer in Stewart Calculus if you've got it

[TrueTears]

yeah the chapter in stewarts on sequence and series is very well set out.