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September 29

For which $\alpha \in \mathbb {R} ^{+}$ does $\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{n}}{n!^{\alpha }}}$ tend to zero for $x\to \infty$ ?

Count Iblis (talk) 00:21, 29 September 2016 (UTC)[reply]

Presumably you mean for x ∈ ℝ (as opposed to ℂ), and that it tends specifically to positive infinity. You could also include α = 0 if you permitted analytic continuation. It is easy to find some values (0, 1, 2) of solutions of α, but I suppose you're not interested in isolated solutions. —Quondum 03:37, 29 September 2016 (UTC)[reply]

Yes, 0, 1, 2, yield 1/(1+x), exp(-x) and J0(2x) respectively, so I'm interested in the general case. Count Iblis (talk) 21:47, 29 September 2016 (UTC)[reply]

For

α = 2

you seem to have mistyped the expression

J 0 (2 \sqrt x)

. For larger integer values (3, 4, etc.), the limit seems to be

\infty

. So a strawman hypothesis (considering the numeric observation below) might be that there is a range

0 \leq α \leq ω

for which the limit is zero, and above the boundary

ω

it is infinite, where

2 \leq ω \leq 3

. The series is clearly convergent for all finite

x

with

α > 0

. I can't help feeling that it should not be difficult to find

ω

, or at least a moderately good upper bound on it, after which one would show that it definitely goes to infinity due to the fast convergence, though I don't know how to formulate that. —Quondum 06:29, 30 September 2016 (UTC)[reply]

Maybe more likely oscillatory than infinite? Plotting out a bit for alpha = 3 suggested that it wiggles pretty vigorously for large x. --JBL (talk) 19:20, 5 October 2016 (UTC)[reply]

Numerically it seems to go to zero for all

0\leq \alpha \approx \leq 1.75

. 24.255.17.182 (talk) 05:01, 29 September 2016 (UTC)[reply]

Crocodile problem: are all these solutions right and acceptable in an exam?

This problem was widely reported by several media, some even claiming it was unsolvable. A description of the problem can be found here: [1].

I wonder whether there are several ways of solving it. All three solutions below lead to the same value, 8m, one is simple trial-and-error, one is trigonometric, and the other derivative. In the exam, at no point there was any indication about what methods to use.

Trial-and-error: plug integer values in the equation, once you found the lowest (that is, 8m) try values around it, that is 7.999999 or 8.000001. That will assure you a certain level of precision, but it's obvious that maybe it could still be a number with many decimal places like 8.00000000001.

Trigonometric: calculate speed in land and water. That's a pace of 4 and 5 tenth of second per meter respectively. The river is 6m wide. Going over land/going through water are in a relationship of 4/5. That is, 4 on land is equal to 5 through water. Divide 4/5 to obtain 0.8, which is the sine of the angle 53.13. Get the cosine of the same angle, that's 0.6, 6/h = . 0.600. The hypotenuse is 10m. x is, per Pythagoras theorem, 8m.

Derivative:

$T(x)=5*(36+x^{2})^{1/2}+(80-4x)$

$T'(x)=5/2*(36+x^{2})^{-1/2}*2x-4$

$T'(x)={\frac {5x}{\sqrt {36+x^{2}}}}-4$

${\frac {5x}{\sqrt {36+x^{2}}}}-4=0$

$5x=4*{\sqrt {36+x^{2}}}$

$25x^{2}=16*(36+x^{2})$

$25x^{2}=576+16x^{2}$

$x^{2}=64$

$x=8$

Llaanngg (talk) 19:46, 29 September 2016 (UTC)[reply]

I've edited the derivative calculation to remove about six typos that presumably occurred when translating into wikitext. --69.159.61.230 (talk) 22:19, 29 September 2016 (UTC)[reply]

The problem looks like a test of calculus skills. Ruslik_Zero 20:13, 29 September 2016 (UTC)[reply]

Yes, it seems clear that the derivative method is the intended solution and is valid. It identifies a local extremum and from consideration of other points on the (x,T) curve that must be a minimum. As to trial-and-error, Llaanngg himself or herself has explained why it is not a mathematically valid solution.

The trigonometric method is interesting, but is based on an unstated and unproven premise. This solution involves a right triangle where one side is along the river bank (on land) and the hypotenuse is the diagonal that the crocodile swims (in water). The unstated premise is that the overall time is minimized if these two sides are in proportion such that the side on land is to the side in water as the speed in water is to the speed on land. Generalizing the problem by substituting arbitrary constants for the numerical ones, I see that this does indeed always produce the right result, but I have to say it is not obvious to me why it should, so I think some proof of that premise is required.

--69.159.61.230 (talk) 22:19, 29 September 2016 (UTC)[reply]

This is a very old problem. A similar problem appears in chapter 26 of The Feynman Lectures on Physics, as the problem of a person on shore trying to reach a drowning girl in the minimum amount of time. (He uses it as an analogy for the refraction of light passing between two media with different indices of refraction). The problem as presented in the link above is almost trivial, since they hand you the equation relating time to the shoreline position, instead of the usual presentation which just gives you the speed on land and the speed on water and lets you work out the equation yourself. It's pretty sad comment on the level of mathematical literacy in society if some media are claiming it is unsolvable, or even particularly difficult. CodeTalker (talk) 23:26, 29 September 2016 (UTC)[reply]

I would think that the expected answer would probably have been the one involving calculus, though I suppose you might get credit for the trigonometric one if you proved the unstated assumption that it makes. Double sharp (talk) 01:37, 30 September 2016 (UTC)[reply]

The simplest method, by far, is to use a graphing calculator and take the minimum (8,98). While an advanced class may not allow this, it's a good way to solve the problem in a basic math class. That is, it shows the student knows how to use the tools available to solve problems. StuRat (talk) 15:52, 1 October 2016 (UTC)[reply]

Actually I daresay it would be the advanced class that would allow it, and the basic class that would not: the idea being that you're only allowed to use the tools once you are very clear about what you're actually doing with them, and you know how you could do it yourself if the situation arose. Double sharp (talk) 15:59, 1 October 2016 (UTC)[reply]

You might be right, but we do need to change the way education is done. For people who aren't math or science majors, it's a waste of time and resources to try to teach them techniques they won't use in their jobs, and will soon forget. What's more important, for those students, is to teach them how to access the available info and use the tools available to them, to solve real world problems.

To reverse the situation, it may be important for an arts major to memorize many artists and their works, but is it critical for math and science majors to do so ? If they ever need to know who wrote a particular sonata, they can just Google it. StuRat (talk) 16:06, 1 October 2016 (UTC)[reply]

Education is not only (perhaps not even primarily) about job preparation, nor should it be. (Not that this tangent has anything to do with the question asked ....) --JBL (talk) 04:43, 2 October 2016 (UTC)[reply]

There are some basics which everyone should know, such as how to read and how their government works, but calculus isn't in that category. As far as math goes, I'd say the typical non-STEM citizen needs nothing beyond the basics of how interest rates and compounding work. StuRat (talk) 19:27, 5 October 2016 (UTC) [reply]

For an opposing point, see Robert A. Heinlein: Anyone who cannot cope with mathematics is not fully human. At best, he is a tolerable subhuman who has learned to wear his shoes, bathe, and not make messes in the house. Personally, I would not state it quite as extremely, but I do think that the typical citizen needs to understand things like the square-cube law, exponential growth (of which compounding is only one aspect), and basic statistics. How else is he to evaluate different statements and models to make informed personal and political decisions? --Stephan Schulz (talk) 21:46, 5 October 2016 (UTC)[reply]

September 29

For which α ∈ R + {\displaystyle \alpha \in \mathbb {R} ^{+}} does ∑ n = 0 ∞ ( − 1 ) n x n n ! α {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{n}}{n!^{\alpha }}}} tend to zero for x → ∞ {\displaystyle x\to \infty } ?

Crocodile problem: are all these solutions right and acceptable in an exam?

For which $\alpha \in \mathbb {R} ^{+}$ does $\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{n}}{n!^{\alpha }}}$ tend to zero for $x\to \infty$ ?