Wikipedia:Reference desk/Archives/Mathematics/2013 August 26

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August 26

Values of the arcsecant function

I'm teaching a calculus course from Stewart's textbook. He defines Arcsec as taking values in [0,π/2) ∪ [π,3π/2). I would naturally have preferred [0,π/2) ∪ (π/2,π], since Arcsec(x) = Arccos(1/x) seems a highly desirable relation to have.

What advantage do people like Stewart see in adopting the first convention?

Is there a consensus for one or the other in more rigorous math texts?

Thanks. — Preceding unsigned comment added by 96.46.197.111 (talk) 00:08, 26 August 2013 (UTC)

I happen to own Stewart's book, but my other calculus text (Bers) uses your definition. Probably the main reason for Stewart's convention is you don't need an absolute value in the the derivative. My, next most rigorous analysis text (Protter and Morrey) doesn't cover it. I doubt there's an accepted standard and, imo, it's not the kind of thing you want students obsessing over anyway. --RDBury (talk) 06:49, 26 August 2013 (UTC)

I had never thought about the derivative, but you're right, the formula is simpler that way. It's still strange. And I never thought for a second that my students would obsess over it! Not mine, anyway. Thanks for the answer. 96.46.200.152 (talk) 06:02, 28 August 2013 (UTC)

Limit of an (infinite) sum, and teaching a man how to fish

1. What is the limit of the sum of (2/3)*(1/5) + (2/3)*(4/5)*(1/7) + (2/3)*(4/5)*(6/7)*(1/11) + ... + (2/3)*...*(1/Pm), as the prime Pm approaches infinity? I had reason to think that the limit was 1/3+function(Pm), but the ratio test gives (Pm-1)/Pn (Pn being the prime following Pm), which looks very much like Pm/Pn, the ratio test for the divergent sum of inverses of primes.

2. My college courses that touched on infinite series dealt much more with tests proving whether or not a series converged, as opposed to finding the actual convergence limit (aside from simple stuff like geometric series). Could anyone be so kind as to provide a list of FREE places, whether in Wikipedia or elsewhere, that give/teach the rules and tricks for calculating the convergence limits of infinite series. (This way I can stop asking for help every time I encounter an infinite series.) TIA. — Preceding unsigned comment added by Bh12 (talk • contribs) 12:18, 26 August 2013 (UTC)

For (2), try Paul's Online Calculus Notes here [1] OldTimeNESter (talk) 20:07, 26 August 2013 (UTC).

If you're looking for info on convergence tests on wikipedia, try [2] and Convergence tests.Phoenixia1177 (talk) 21:24, 26 August 2013 (UTC)

On your point 2 — there aren't any rules for figuring out in general what an infinite sum, if it converges, adds up to. Much of the time there won't be any way of writing it that's simpler than the infinite sum itself. That's (at least part of) why your classes concentrated more on convergence tests — it's much more likely that you can actually get an answer.

However, sure, there are various tricks that sometimes work. See generating function for one rather nice one. --Trovatore (talk) 21:35, 26 August 2013 (UTC)

As for 1, the general term is asymptotic to $(\log(p_{n})p_{n})^{-1}$, which is not summable. Sławomir Biały (talk) 23:45, 26 August 2013 (UTC)

1. I misunderstood the results of the ratio test; when the limit is 1 then it tells you nothing. But knowing that the general term is asymptotic to 1/(Pn*ln(Pn)) may be of help, even if it is not summmable - thank you.

2. Sigh. I read Paul's Online Notes and it more or less says the same thing: it's difficult and there aren't any rules for figring it out in general. Bh12 (talk) 00:29, 27 August 2013 (UTC)

Well, you know, welcome to mathematics. If there were rules for figuring it out in general, it wouldn't be interesting — we'd just program them into a computer once and for all, and be done with it. --Trovatore (talk) 21:08, 27 August 2013 (UTC)

HOW do you know that the general term is asymptotic to 1/(P*ln(P))? Not that I doubt you, but what's the proof? Some book or paper? Which one?

Regarding "tricks", I read the Wikipedia article titled Generating Functions, but could not figure out that "rather nice one" - what is it?

I am open to hearing more about various rules/tricks for determining the sums, partial sums. and general terms of series, including getting links to sources. Bh12 (talk) 10:53, 27 August 2013 (UTC)

Use the fact that $\sum _{k=1}^{n}{\frac {1}{p_{k}}}\sim \log \log p_{n}.$ Then applying Taylor's theorem for the logarithm gives $\log \prod _{k=1}^{n}\left(1-{\frac {1}{p_{k}}}\right)\sim -\log \log p_{n}$. Exponentiating both sides gives the estimate that I referred to. Sławomir Biały (talk) 11:44, 27 August 2013 (UTC)

1. Back to the orginal question: Is that infinite sum ((2/3)*(4/5)*(6/7)*(10/11)*...*(1/p)) convergent, and according to which test?

2. Thank you for the steps of the proof for 1/(p*ln(p)). Bh12 (talk) 20:27, 27 August 2013 (UTC)

Sorry, I was hasty about my conclusion. It does converge, since $p_{n}\sim n\log n$ and so the nth term of the series is asymptotic to $(n\log(n)^{2})^{-1}$ which is summable (e.g., by the Cauchy condensation test). Sławomir Biały (talk) 01:53, 28 August 2013 (UTC)

Great!! One (last?) question: $(n\log(n)^{2})^{-1}$ means 1/(n*(log(n))*(log(n))), no?Bh12 (talk) 14:12, 28 August 2013 (UTC)

Yes, right Sławomir Biały (talk) 14:20, 28 August 2013 (UTC)

Okay, one more question. The infinite sum of 1/(n^2) is a bit more than 1.5, or (pi^2)/6 exactly. How is this vlaue of (pi^2)/6 obtained?

(By shifting the 1-wide rectangles of 1/(n^2) to left by 1/2, then the curve of y vs. 1/(x^2) passes through the middle of the top of each rectangle, giving (at least) 1.5, and due to the curve's concavity the sum is a bit more than 1.5. But the value (pi^2)/6 eludes me.)Bh12 (talk) 13:10, 30 August 2013 (UTC)

We have an entire article on this question: Basel problem.—Emil J. 14:47, 30 August 2013 (UTC)

Definition of Division

My textbook has the following definition: "If a and b are integers with a ≠ 0, we say that a divides b if there is an integer c such that b = ac." It does not say that c > 0. Is this an error? And if not, is there ever a reason to define c < 0, since for -b we can write the more intuitive -b = (-a)c? (That is, -b equals -a times c?) OldTimeNESter (talk) 17:25, 26 August 2013 (UTC)

The two definitions are equivalent, as you noted. (Assuming you meant c ≥ 0 rather than c > 0, since any integer a > 0 divides 0.) Therefore, omitting the condition is not an error. The desired behavior is to have -2 | 4 in most cases when dealing with integers that are allowed to be negative. In fact, as noted in our article Divisor, some authors also omit the a ≠ 0 condition, in which case 0 divides 0 but not any other number. « Aaron Rotenberg « Talk « 18:36, 26 August 2013 (UTC)

Calculus - Function describing path of a car's back wheels.

I have been working several months on a question, but have not been able to find an answer. If a car starts driving with its front wheels turned as far as possible in one direction, but the steering is adjusted so that the front wheels follow a straight path, what mathematical function will describe the path of the rear wheels?

So far, I have created a sketch with a theoretical car whose front wheels can turn 90º. It is facing "south" on the y axis, with its front wheels on the x axis facing "east". The wheelbase is represented by the constant k.

At any time, the rear wheels are moving directly towards the front wheels. I came up with an equation to represent the line between the front and rear wheels, which is always tangent to the path of the rear wheels.

I have represented this situation with an equation that sets the slope of the tangent line equal to the derivative of f(x). Using trigonometry, I can algebraically represent the horizontal distance between the wheels, and the vertical distance between the wheels is the value of f(x).

Here is the equation I obtained: f'(x) = -√(k^2 - y^2) / y

My instinct is to find the integral of both sides of this, but once I have done so (using WolframAlpha), the resulting function does not seem to be what I am looking for. I would be happy for someone to discuss this with.

Hubby2debbie (talk) 19:17, 26 August 2013 (UTC)

I don't know the answer, but I can give you some links to head in the right direction. Really, it depends on how much detail you want in your model. See tractrix for an idea of how to deal with a point being dragged by a string. This could probably be adapted to treat e.g. a bicycle (with the back wheel always traveling toward the front, but the position of the front can be controlled). But, in general, a four-wheel car with only two front steering wheels constitutes a Nonholonomic system, and its very tricky to work out paths of component parts. See also Parallel parking problem. SemanticMantis (talk) 20:48, 26 August 2013 (UTC)

1) "At any time, the rear wheels are moving directly towards the front wheels." This doesn't seem right to me.

2) The problem with representing this mathematically is that the rear wheels sometimes slip a bit, especially in tight turns. If they make a squealing sound, this is probably why. The tires don't have zero thickness in contact with the ground, so this means during turns one side of each tire must travel further than the other. (I've often thought a whole bunch of skinny tires which can rotate at different speeds might work better than only 4 thick tires.) StuRat (talk) 01:42, 27 August 2013 (UTC)

I don't think your premises make sense. The car's first-order motion can be described as rotation around some point to the left/right of the car. If the wheels aren't slipping, then every line passing through a wheel and orthogonal to it will pass through the point of rotation. So if you think about what would happen if both the front wheels face the same direction, it wouldn't work out.

In any case (even if the wheels do slip), it the car's motion is at some steady state, it can still be described as rotation around a point. So for each will, the path will be $(x,y)=(x_{0}+r\cos(\omega t+\varphi ),y_{0}+r\sin(\omega t+\varphi ))$, where $(x_{0},y_{0})$ is the center of rotation, $\omega$ is the angular velocity, and $r$ and $\varphi$ are the distance from the center and phase for the particular wheel. -- Meni Rosenfeld (talk) 18:36, 1 September 2013 (UTC)

Axioms for arithmetic

I'm confused by something i just read in Joseph Mazur's book Euclid in the Rainforest (pub. 2006): "Natural numbers have been used correctly for thousands of years before anyone thought of analyzing them. Only in the past century have we felt the need to find axioms for arithmetic. The superstructure of almost all of mathematics was not compromised by the lack of understanding of its core foundations." (p.90) What exactly is he referring to in the second sentence? I've looked over the wiki articles on number theory but can't find an answer. A Google search for "axioms for arithmetic" brings up only Peano axioms from the 19th century. Likewise, analytic number theory seems to have begun in the 19C. Any help is appreciated. 76.17.125.137 (talk) 20:19, 26 August 2013 (UTC)

I don't know the book, but he is probably referring to the Peano axioms, formulated in 1889, about 124 years ago. Perhaps the author was being imprecise in his date arithmetic? --Mark viking (talk) 20:36, 26 August 2013 (UTC)

Another possibility: Mazur was referencing Principia_Mathematica, and/or ZFC, both of which were slightly less than 100 years old when Mazur's book came out. My understanding is that, to Bertrand Russell et al., Peano's axioms as previously written did not constitute an iron-clad, axiomatic, rigorous development of basic arithmetic. SemanticMantis (talk) 20:42, 26 August 2013 (UTC)