Warwick First Year Maths-Past paper questions

Ok here is my fail attempt at 2006 analysis

I may improve on this in the morning

Reply 83

benwellsday

OP

The first few answers for (1) don't seem to match the questions. Not sure what happened there. Actually, nothing matches 2006. I think this is 2007.

Here's anything I've got to say anyway

Reply 84

benwellsday

The first few answers for (1) don't seem to match the questions. Not sure what happened there. Actually, nothing matches 2006. I think this is 2007.

Tom told me to do 2006, but I was very drunk and didn't notice I was actually doing 2007.

benwellsday

And I didn't do (d) on paper, so no ideas there.

No matter. Thanks for your help :smile:

Reply 85

hey guys,
how do you prove from the completeness axiom that an increasing sequence an which is bounded above is convergent.

do you start with saying its non-empty and bounded above so there's a least upper bound, say U.

then a(n) is less than or equal to U for all n.

then for all positive epilson there's N s.t
la(n) - Ul < epilson for all n>N

but it seems like i'm fudging it to get epilson into this. i've lost workbook 5, and i think it's from there.

thanks for any help, i just want a rigorous answer.

Reply 86

Ok, the sequence isn't what's non empty. The set of the members of sequence is non empty, so it has a supremum because of the completeness axiom.

I've written an answer above per Q3 from the part which begins 'lemma'. Benswellsday points out where the completeness axiom is used in the comments related to the question

Reply 87

thanks, so you have to definietely prove the lemma as well for the proof or can you just use it?

Reply 88

i'm doing 2005 question 4a. for analysis. can someone please quickly compare answers.

first, is convergent as less than 1/(n^2) and is absolutely convergent too.

2. is bigger than 1/n so diverges by comparison test to +infinity

3.converges by alternating series test, and converges conditionally ( as n^-.5 doesn't converge)

4.converges absolutely ( as equal to 2^-n)

Is what i'm writing what you would write in an exam or for this question would you have to show how you figured it out???

thanks in advance guys

Reply 89

amo17

thanks, so you have to definietely prove the lemma as well for the proof or can you just use it?

Because it makes up most of the proof, I think you have to prove it.

Reply 90

amo17

i'm doing 2005 question 4a. for analysis. can someone please quickly compare answers.

first, is convergent as less than 1/(n^2) and is absolutely convergent too.

2. is bigger than 1/n so diverges by comparison test to +infinity

3.converges by alternating series test, and converges conditionally

4.converges absolutely

Is what i'm writing what you would write in an exam or for this question would you have to show how you figured it out???

thanks in advance guys

I agree with those answers, though you might want to say what test you used in part 4. Also, you might want to say why 3 doesn't converge absolutely.

Reply 91

thanks irrational number!

is my reasoning sound now?

i'm lost as what to do for question 4 part c. any hints???(lol there's isn't enough for me!)

thanks again. :smile:

Reply 92

You might not want to write things like 'as n^-0.5 doesn't converge', since it's not true. The sum doesn't converge.
Also, I think your reasoning is wrong for iv). It's not n!/2(n!) it's n!/(2n)!.
I haven't looked at part C) and I'm going out in a minute. I'll have a crack when I get back if someone else hasn't already and I'm sober enough...

Reply 93

lol, k thanks for your help.

what else could you do for iv.?

Reply 94

Well usually when it involves factorails in sums.... I would try the ratio test

Reply 95

oh ok, thanks so the ratio test goes to 1/2(2n+1), which converges to zero, so the sum converges to 0.

would they ever ask us to work out what the sums actually converge to or just if they converge, do you think?

Reply 96

k guys, i have another question but i've attached it cos i can't write it out clearly. is the answer N=1000?

cheers

amo17

k guys, i have another question but i've attached it cos i can't write it out clearly. is the answer N=1000?

cheers

When faced with a similar question in my real exam, I wrote N=420^456723287329 or something ridiculously high, because I couldn't be assed to do the algebra for two marks. They don't ask you to prove it after all, you just have to spot that (3n+1)/n converges to 3 :p:

. I amused myself by adding more digits onto the exponent whenever I got stuck elsewhere.

Reply 98

JohnnySPal

When faced with a similar question in my real exam, I wrote N=420^456723287329 or something ridiculously high, because I couldn't be assed to do the algebra for two marks. They don't ask you to prove it after all, you just have to spot that (3n+1)/n converges to 3 :p:

. I amused myself by adding more digits onto the exponent whenever I got stuck elsewhere.

thanks ok, i will take that advice. but seriously out of interest i want to know how to do the question. is it just a rearrangement?

thanks

Reply 99