# An Interesting Problem

Discussion in 'Physics & Math' started by §outh§tar, Jul 1, 2008.

1. ### Guest254Valued Senior Member

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1,056
But you can - you only need to know how to expand log(1+x) for small x...
That's pretty much the definition of a power series... (or, more formally, an asymptotic series). What I've written is perfectly rigorous.

3. ### Guest254Valued Senior Member

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1,056
Excellent, well done. This is actually something I've given in interviews, as it's easier to construct an argument by conversing, than it is to actually write down. I.e, one says "well, there can't be a 5 because 5=2+3, and 2x3=6>5 etc...

5. ### §outh§taris feeling causticRegistered Senior Member

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If a, b > 0, find $S_p (a,b) = \left( {\frac{{a^p + b^p }}{2}} \right)^{\frac{1}{p}}$
as p tends to +infinity and as p tends to -infinity

7. ### CptBorkValued Senior Member

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If $a\geq b$, then the limit is $a$ as $p\mapsto\infty$ and the limit is $b$ as $p\mapsto -\infty$.

8. ### §outh§taris feeling causticRegistered Senior Member

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Correct, CptBork. We'd like a proof though.

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What is the largest number of coins from which one can detect a counterfeit in three weighings with a pan balance, if it known only that the counterfeit coin differs in weight from the others?

9. ### CptBorkValued Senior Member

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Well I'm too lazy to type out the whole entire argument, I did write it out with pen and paper though. Start with $a\geq b$. Then we write:

$\left(\frac{a^p+b^p}{2}\right)^{1/p}=a\left(\frac{1+\left(\frac{b}{a}\right)^p}{2} \right)^{1/p}$

Next we write

$\left(\frac{1}{2}\right)^{1/p}\leq\left(\frac{1+\left(\frac{b}{a}\right)^p}{2}\right)^{1/p}\leq1$

and apply the squeeze theorem as $p\mapsto\infty$. That takes care of the first part. Second part, $p\mapsto-\infty$, is similar and just requires a little extra manipulation.

10. ### §outh§taris feeling causticRegistered Senior Member

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Show that if $\frac{{b_n }}{{b_{n + 1} }} = 1 + \beta _n$, n=1,2,..., and the series $\sum\limits_{n = 1}^\infty {\beta _n }$ converges absolutely, then the limit ${\lim }\limits_{n \to \infty } b_n$ exists.

11. ### §outh§taris feeling causticRegistered Senior Member

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Show that if p is a real number then the series with nth term $\frac{{(n + p)n^p - (n + 1)^p n}}{{(n + 1)^p n}}$ converges.

12. ### §outh§taris feeling causticRegistered Senior Member

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4,832
Show that if $a_n > 0$ and ${\lim }\limits_{n \to \infty } \frac{{a_{n + 1} }}{{a_n }}$ exists then ${\lim }\limits_{n \to \infty } \frac{{a_{n + 1} }}{{a_n }} = {\lim }\limits_{n \to \infty } \sqrt[n]{{a_n }}$

In particular, if the left hand limit exists in the following then the right hand limit exists and equality holds: ${\lim }\limits_{n \to \infty } a_{n + 1} - a_n = {\lim }\limits_{n \to \infty } \frac{{a_n }}{n}$

13. ### CptBorkValued Senior Member

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I'm really glad you asked this question. These are some of the very results I learned in analysis class and were then used to provide clever proofs for other results. I've been doing some thinking on the pennies problem so I'll get back to you later.

14. ### CptBorkValued Senior Member

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Mind you, I should have said analysis classes, in plural, since I've taken 4 of them to date. But I always found the first 2 I took to be the most interesting and direct, after that it was all metric spaces and measure theory and abstract stuff.

Anyhow, I did some thinking on the coins problem and I believe the answer is that if you're allowed up to 3 weighings, you can guarantee finding the counterfeit coin in 3 or less measurements if you have 8 or less coins to start with. If you start with 9 or more coins, then there's always a chance that you will fail to detect the counterfeit in 3 measurements or less no matter what algorithm you use. I can post my reasoning if you want, but first I want to check if I have the right answer. My method was to treat it on a case by case basis, considering the worst possible cases for 3, 4, 5 coins etc. until you get up to 9 or more coins.

15. ### §outh§taris feeling causticRegistered Senior Member

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CptBork, I'll have you know that I don't know what "the" answer is to the coins problem. The book I got the problem from doesn't come with answers, not to mention some of its questions are known to be completely wrong..

16. ### §outh§taris feeling causticRegistered Senior Member

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What is the smallest number of questions to be answered "yes" or "no" that one must pose in order to be sure of determining a 7-digit telephone number?

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I don't mean to sidetrack anyone from the current problem, but I have a question regarding SouthStar's second problem. The question was:

Evaluate the limits

$\lim_{x\to0}\frac{\ln\cos x}{(\sin x)^2}$.

Additionally, on July 3, SouthStar wrote:

I know I haven't looked at problems similar to this in a while, but I don't quite understand from where this came. Did you apply l'Hopital's Rule?

18. ### VkothiiBannedBanned

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That looks like a standard rearrangement of trig functions, to me. You know what cos(x), or cos(2x) can be substituted with?

19. ### §outh§taris feeling causticRegistered Senior Member

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Amadeusboy, the second equality comes from standard properties of a*ln x = ln (x^a) and (cos x)^2 = 1- (sin x)^2. The third comes from a use of power series. Is that what you have trouble with?

(The point of my solution was that it involved (almost) no calculus (depending on how you arrive at sin x ~ x as x -> 0))

20. ### ReikuBannedBanned

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In other words, you proved his point.

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I don't believe any trigonometric substitutions were made. Even if they were, what happened to the $\ln$ term in the third equality?

After more work and refreshing my memory of l'Hopital's rule, my approach was as follows:

Let $f(x)=\ln(\cos x)$ and $g(x)=\sin^2 x=(\sin x)^2$.

Then

$f^\prime (x)=-\frac {\sin x}{\cos x}$,

and

$g^\prime (x)=2 \sin x \cos x$.

Accordingly,

$\Large\frac {f^\prime (x)}{g^\prime (x)}=\frac{-\frac{\sin x}{\cos x}}{2\sin x\cos x}=-\frac{1}{2\cos^2x}$.

Thus

$\Large\lim_{x\to0}\frac{f^\prime (x)}{g^\prime (x)}=\lim_{x\to0}\frac{-1}{2\cos x}=-\frac{1}{2}$.

If my line of reasoning is incorrect or I have made a mathematical error, please let me know.

(SouthStar, I didn't see your last post before I finished writing and posting mine. I looked at the problem again using your hints and I understand your approach. Thanks for your response. Amadeusboy)

Last edited: Jul 8, 2008
22. ### ReikuBannedBanned

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Looks good to me...

23. ### CptBorkValued Senior Member

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And what exactly would that point be? I never got around to answering the question yet.