Student 999
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#1
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#1
Struggling to show the factorial part of the expression, is my working out in showing adequate or is there a cleaner way to go about showing it?Name:  Screenshot 2022-07-03 at 23.49.36.png
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Last edited by Student 999; 1 month ago
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RDKGames
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(Original post by Student 999)
Struggling to show the factorial part of the expression, is my working out in showing adequate or is there a cleaner way to go about showing it?Name:  Screenshot 2022-07-03 at 23.49.36.png
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Top-right looks to be on the right lines.

Notice the following:

\displaystyle \ln(1^1 \cdot 2^2 \cdot 3^3 \cdots (n-1)^{n-1} \cdot n^n) = \sum_{i=1}^n i \ln(i)

So work from there. You want to eventually make use of the fact that

\displaystyle \int_0^1 f(x) \ dx = \lim_{n\to\infty} \sum_{i=1}^n \dfrac{1}{n} f\left(\frac{i}{n}\right)

but with f(x) = x\ln x
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DFranklin
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#3
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(Original post by RDKGames)
\displaystyle \ln(1^1 \cdot 2^2 \cdot 3^3 \cdots (n-1)^{n-1} \cdot n^n) = \sum_{i=1}^n i \ln(i)

So work from there.
To add a semi-hint. It is not "obvious", but it's not actually that hard to get the LHS to look like the \ln\left(\dfrac{n!}{0!}\right) + ... + \ln\left(\dfrac{n!}{(n-1)!}\right) expression.

Spoiler:
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You can literally factor it out; I'd start with n!, then n!/2! and so on. You do want to do a couple of the steps to justify jumping to the final factorization.
Last edited by DFranklin; 1 month ago
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Student 999
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#4
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(Original post by DFranklin)
To add a semi-hint. It is not "obvious", but it's not actually that hard to get the LHS to look like the \ln\left(\dfrac{n!}{0!}\right) + ... + \ln\left(\dfrac{n!}{0!}\right) expression.

Spoiler:
Show
You can literally factor it out; I'd start with n!, then n!/2! and so on. You do want to do a couple of the steps to justify jumping to the final factorization.
Bit confused by what you mean to factor out, are you referring to
1x2x3...(n-1)xn = n!
1x2x3...(n-1)xn = n!/1!
3x4...(n-1)xn = n!/2!
...

hence the product of the above is ln (n! x n!/1! x n!/2! ... n!/(n-1)! ) which can be split into ln (n!/0! ) +
ln ( n!/1!) ... ln (n!/(n-1)!)

but this is essentially what I've already done in the above post if I made it a bit more clearer?
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DFranklin
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(Original post by Student 999)
Bit confused by what you mean to factor out, are you referring to
1x2x3...(n-1)xn = n!
1x2x3...(n-1)xn = n!/1!
3x4...(n-1)xn = n!/2!
...

hence the product of the above is ln (n! x n!/1! x n!/2! ... n!/(n-1)! ) which can be split into ln (n!/0! ) +
ln ( n!/1!) ... ln (n!/(n-1)!)

but this is essentially what I've already done in the above post if I made it a bit more clearer?
I have to be honest, I hadn't looked that far down - I was trusting when mqb said "top right seems to be along the right lines".

So, have you shown the result - um, maybe? You haven't actually "closed the gap"; you haven't even explictly said "this equals the expression in square brackets" (or words to that effect).

I'll be blunt: you really need to think a bit more about how you want to present your work. You're consistently on that fine line where an examiner might be able to work out what you meant and if it you actually knew how to prove the result, but they might easily decide "actually, with the amount of gaps I'm filling in myself I'm basically proving this for Student 999 - that's not OK".

It's often also just hard to puzzle out what you mean: the stuff you write e to work out what you meant, but it's going to take them ages, and if you're unlucky they're just not going to bother. I mean, look at your first line:

\displaystyle \dfrac{x_n \ln\left(\frac{1}{x_n}\right)} {n} \text{ where } n\ge 0 \text{ where }x_0 = 0

My immediate reaction is WTF is x_n? And even if I knew what x_n is, why have you written this expression down? And why has a sum appeared in the next line?

Oh God. I've just realised - you actually wrote this as a LH column going down to the bottom of the page, then restarted in the RH even though your LH column is (a lot) wider than where you started your RH column. I've been reading this in the wrong frickin' order.

So I think I'm going to stop at that point - obviously my initial comment about "closing the gap" is invalid. But the fact that a seasoned mathematician can look at your post and not be able to tell the correct order in which you've laid out your argument speaks for itself.
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Student 999
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#6
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#6
(Original post by DFranklin)
I have to be honest, I hadn't looked that far down - I was trusting when mqb said "top right seems to be along the right lines".

So, have you shown the result - um, maybe? You haven't actually "closed the gap"; you haven't even explictly said "this equals the expression in square brackets" (or words to that effect).

I'll be blunt: you really need to think a bit more about how you want to present your work. You're consistently on that fine line where an examiner might be able to work out what you meant and if it you actually knew how to prove the result, but they might easily decide "actually, with the amount of gaps I'm filling in myself I'm basically proving this for Student 999 - that's not OK".

It's often also just hard to puzzle out what you mean: the stuff you write e to work out what you meant, but it's going to take them ages, and if you're unlucky they're just not going to bother. I mean, look at your first line:

\displaystyle \dfrac{x_n \ln\left(\frac{1}{x_n}\right)} {n} \text{ where } n\ge 0 \text{ where }x_0 = 0

My immediate reaction is WTF is x_n? And even if I knew what x_n is, why have you written this expression down? And why has a sum appeared in the next line?

Oh God. I've just realised - you actually wrote this as a LH column going down to the bottom of the page, then restarted in the RH even though your LH column is (a lot) wider than where you started your RH column. I've been reading this in the wrong frickin' order.

So I think I'm going to stop at that point - obviously my initial comment about "closing the gap" is invalid. But the fact that a seasoned mathematician can look at your post and not be able to tell the correct order in which you've laid out your argument speaks for itself.
Appreciate the critique, I believe my post has came across in the wrong light due to my poor choice of words. I understand that presentation is lacking ,what I was really trying to ask was if the above that I typed up plus some kind of sentence explaining what I had done be sufficient to show the expression in the square brackets or is there some kind of thing they're looking for specifically like an induction proof which is more vigorous I guess.

the x_n thing didn't really make sense, I kind of wrote it out similar to when I did Simpson's rule where they use similar notation and just assumed anyone reading it would understand however will start explaining more of what I do. A better choice of notation would just be a sum sign.

Was too laxed in writing things out last night even though I do have 'unlimited paper' since its an iPad hopefully by the end of summer you'll see an improvement in the way I present things mathematically, in the meantime please do continue assisting me when you have time, thanks
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DFranklin
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#7
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#7
(Original post by Student 999)
Appreciate the critique, I believe my post has came across in the wrong light due to my poor choice of words. I understand that presentation is lacking ,what I was really trying to ask was if the above that I typed up plus some kind of sentence explaining what I had done be sufficient to show the expression in the square brackets or is there some kind of thing they're looking for specifically like an induction proof which is more vigorous I guess.

the x_n thing didn't really make sense, I kind of wrote it out similar to when I did Simpson's rule where they use similar notation and just assumed anyone reading it would understand however will start explaining more of what I do. A better choice of notation would just be a sum sign.

Was too laxed in writing things out last night even though I do have 'unlimited paper' since its an iPad hopefully by the end of summer you'll see an improvement in the way I present things mathematically, in the meantime please do continue assisting me when you have time, thanks
The point is that your presentation is sufficiently bad that it's really not clear what you're trying to show. My previous comment that:

you're consistently on that fine line where an examiner might be able to work out what you meant and if it you actually knew how to prove the result, but they might easily decide "actually, with the amount of gaps I'm filling in myself I'm basically proving this for Student 999 - that's not OK"
still holds.

Explictly, I'm not happy with you going from ln(1x2x3...n x 1x2x3...n x 3x4...n x 4x5...n...) to ln(1x2^2 x ... x n^n). It's not clear how this happens, you have oddities in your product (first 2 terms are 1 x 2 x ... and then the next is 3 x ...) and it's not obvious where your product ends.

With a little thought, you could have laid this out as:
Code:
1    x    2    x    3    x    4    x    ...    x    n
x         2    x    3    x    4    x    ...    x    n
x                   3    x    4    x    ...    x    n
x                             4    x    ...    x    n
...
x                                                   n

=  1^1 x    2^2 x    3^3 x    4^4 x    ...    x   n^n
which would have the conclusion obvious (you could even add some downwards arrows if not limited to text like me).
Last edited by DFranklin; 1 month ago
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