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HSC 2013 MX2 Marathon (archive) (6 Viewers)

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Sy123

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Re: HSC 2013 4U Marathon

I think it should be

Can I do this using Fourier or Taylor series since they use elementary functions and some integration?
yes that is also true.

Yeah sure if you want, just make sure you can prove everything that you need to, and justify things rigorously (such as radius of convergence etc.)
I would prefer a HSC solution though.

EDIT: Also Heroic, I don't like the step 'I'll assume R(x) is a constant' are you able to justify that it is a constant?
 
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RealiseNothing

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Re: HSC 2013 4U Marathon

Suppose we are flipping a biased coin such that:



Where:

is the probability of flipping tails.

is the probability of flipping heads.

is a positive integer.

Find the probability of flipping heads before tails.
 
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Re: HSC 2013 4U Marathon

I quite like the original, mostly because a lot of the others are too advanced for me to understand right now haha.
I like the one using Fourier series but I still don't completely understand it haha

I want to make a 4U level question to compute , don't know if that is possible though.
 
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Re: HSC 2013 4U Marathon

Suppose we are flipping a biased coin such that:



Where:

is the probability of flipping tails.

is the probability of flipping heads.

is a positive integer.

Find the probability of flipping heads before tails.
Seen a question slightly similar to this in 2000 paper. Don't know if any one in the state managed to get it lol
 

RealiseNothing

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Re: HSC 2013 4U Marathon

I like the one using Fourier series but I still don't completely understand it haha

I want to make a 4U level question to compute , don't know if that is possible though.
You can use the exact same method as the question I posted up I'm pretty sure. The only problem is the assumptions don't make the question that rigorous, and it would be tedious.
 

seanieg89

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Re: HSC 2013 4U Marathon

I quite like the original, mostly because a lot of the others are too advanced for me to understand right now haha.
Haha except the ideas needed to make that proof rigorous are deeper than the ideas in most of the other proofs.
 

seanieg89

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Re: HSC 2013 4U Marathon

Haha except the ideas needed to make that proof rigorous are deeper than the ideas in most of the other proofs.
Armed with the knowledge that it does happen to work though it does make for a pretty efficient way of computing zeta(2n) inductively.
 

HeroicPandas

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Re: HSC 2013 4U Marathon

yes that is also true.

Yeah sure if you want, just make sure you can prove everything that you need to, and justify things rigorously (such as radius of convergence etc.)
I would prefer a HSC solution though.

EDIT: Also Heroic, I don't like the step 'I'll assume R(x) is a constant' are you able to justify that it is a constant?
no... i'm afraid not
 

RealiseNothing

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Re: HSC 2013 4U Marathon

not 100% sure about this actually...
 
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seanieg89

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Re: HSC 2013 4U Marathon

It's pretty ironic isn't it.
Haha a bit. Euler had a knack for doing some whacky things on intuition that turned out to be things you could make rigorous. Its a pretty bad habit to get into unless you have the intuition of Euler :p. (He never claimed that the random manipulations comprised a proof I don't think, he just used such magic to convince himself of the plausible truth of statements before he rigorously proved them.)

On the whole the standard of rigour is higher these days as well.
 
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Re: HSC 2013 4U Marathon

You can use the exact same method as the question I posted up I'm pretty sure. The only problem is the assumptions don't make the question that rigorous, and it would be tedious.
hmm I will try to do Apéry's constant then :p

PS: this can't be done, right? Since it doesn't converge to a nice value.
 
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RealiseNothing

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Re: HSC 2013 4U Marathon

Its a bit more complicated than that but I remember that it can be done.
Don't you end up with:

is identical to:



where a,b,c.. are roots of sinx

And equate the co-efficient of 2n for whatever integer value of n you choose?
 

seanieg89

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Re: HSC 2013 4U Marathon

Don't you end up with:

is identical to:



where a,b,c.. are roots of sinx

And equate the co-efficient of 2n for whatever integer value of n you choose?
I don't think so but post your working for zeta(4) if you believe so.
 

RealiseNothing

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Re: HSC 2013 4U Marathon

From the infinite series of sine:



Also:



By difference of two squares:



So we can say that:



is identical to



hence they have equal co-efficients. Equating co-efficients of x^4 gives:



ok I see the problem.
 

Sy123

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Re: HSC 2013 4U Marathon

Suppose we are flipping a biased coin such that:



Where:

is the probability of flipping tails.

is the probability of flipping heads.

is a positive integer.

Find the probability of flipping heads before tails.
The problem simply boils down to computing:





Is it possible to compute



? Its easy if x is 1 but....
 
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