May 08 Paper 3 Series And Differential Equations TZ 2 Question 5 b

[iBS2012]

Hi,

Since I've recently switched to my 3rd math teacher in the course of my IB (!), my entire class is lagging behind quite a bit on the option. I've started looking through some past papers and its going surprisingly well so far, but anyway: I've come across a question from the 2008 paper:

Now my teacher was nice enough to let me take a look at the mark scheme which is along the lines of:

∛(n+1)-n=n(∛(1+1/n³)-1)

=n(1+1/3n³-1/9n⁶+5/81n⁹-...-1)

First question is how they got to this last part?

Going on from here, they used v=1/n² as the "auxillary series". Could someone please explain what this means and what exactly its used for?

I still have problems with some of the later parts of the solution but I think I might be able to figure these out once I get the earlier parts.

Thanks

Good luck to everyone else who's sitting the may session!

Edited April 3, 2012 by iBS2012

[Grassroot]

a. using ratio test, make the (n+1)/n term less than 1. You should be able to obtain the answer

b. using integral test you should be getter the result

I believe the integration in part b is a bit complicated....

sry my respond maybe a bit superficial... PM me if u need the detail

[iBS2012]

I remember the mark scheme saying that you can only get 3 out of 7 marks for using the integral test.

And check your inbox (:

[Grassroot]

I remember the mark scheme saying that you can only get 3 out of 7 marks for using the integral test.

And check your inbox (:

just checked MS.... I admit my answer is not correct.... seems like MS is using approximation i.e. taylor expansion. to solve this problem....

I bet I won't be able to solve this in the exam...luckily my option is stat but I do have a large opportunity to see such question in FM paper

[iBS2012]

Yea, that was my thought, but how they expanded that series is honestly beyond me, especially with the mark allocation where it is.

Anyone else have any insights on this question?

[Grassroot]

Yea, that was my thought, but how they expanded that series is honestly beyond me, especially with the mark allocation where it is.

Anyone else have any insights on this question?

That is just a normal expansion which you may find it in the textbook... actually what surprised me is the mentioning of the approximation rather than approximation itself

[Dinstruction]

If you take the limit of the quotient of two series as n approaches infinity, and the limit is finite and positive, then both series either converge or diverge.