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Not entirely is the operative word. Simplexes are real concepts. No.

You made a mistake in deriving the third derivative. That led you to the wrong pattern in the nth derivative of fn+1(x). Seeing tentative patterns and then proving them to be true is how a lot of the recursive-type relations were developed.

No. Second derivative means [math]\frac {d^2}{dx^2}[/math]. The equation in post number 14 is the first derivative, so differentiating again will yield the second derivative.

I already showed you the first derivative, a simple application of the chain rule: [math]\frac{d}{dx}f_{n+1}(x) = x\frac{d}{dx}f_n(x) + f_n(x)[/math] Differentiate the above to get the second derivative of fn+1(x).

Try it one step at a time. What is the second derivative, the third, ... You should see a pattern. Then prove that the pattern is general.

What you do now is find a way to express [math]\frac{d^{n+1}}{dx^{n+1}}f_{n+1}(x)[/math] in terms of [math]f_n(x)[/math] Hint: [math]f_{n+1}(x) = x f_n(x)[/math] Differentiating, [math]\frac{d}{dx}f_{n+1}(x) = x\frac{d}{dx}f_n(x) + f_n(x)[/math] What happens when you differentiate again, and again, and ... n+1 times? Can you see a pattern arising?

No. Why don't you at least try doing what I suggested in posts #2 and 4?

You are ignoring that the functions being differentiated on the left-hand side are different for each n. In particular, this statement is wrong: "so when we do another derivative we should get the n=k+1" Please type your posts! I had to type your response by hand because there is no way to quote or cut-and-paste text from an image.

I disagree. Mathematics is largely a human invention, motivated by reality. Mathematical discovery involves pulling stuff out of the invented axioms.

Re LaTeX: See this Quick LaTeX Tutorial thread. Apparently MathType can export LaTex. See http://www.dessci.com/en/support/mathtype/workswith/t/latexee.htm. You didn't do anything close to what I suggested you do. I said to forget about the right-hand side for a bit. The problem is that you are ignoring that the functions being differentiated are themselves a series. Hint: Given only that [math]f_{n+1}(x) = xf_n(x)[/math], what is [math]\frac{d^{\,k}}{dx^k}f_{n+1}(x)[/math]?

Please learn to use this forum's [math] capability. That way I can quote your work and help you along the way. You didn't get the right answer because you solved the wrong problem. The problem at hand is to show [math]\frac{d^n}{dx^n}\left(x^{n-1}e^{1/x}\right) = (-1)^n\frac{e^{1/x}}{x^{n+1}}[/math] Define some terms to make this a bit easier to understand: Let [math]f_n(x) \equiv x^{n-1}e^{1/x}[/math] [math]g_n(x) \equiv (-1)^n\frac{e^{1/x}}{x^{n+1}}[/math] With these definitions, the problem is [math]\frac{d^n}{dx^n}f_n(x) = g_n(x)[/math] Hint: Forget about the right-hand side for a bit. First find a way to express [math]\frac{d^{n+1}}{dx^{n+1}}f_{n+1}(x)[/math]. Second hint: [math]f_{n+1}(x) = xf_n(x)[/math]

Ah yes, those lush tropical Texas rainforests. The report attributes to cattle raising *all* of the deforestation in Brazil, Africa, China, and elsewhere. Couple that with exaggerations of how well forests and how poorly croplands sequester CO2 and, voila! "9% of all anthropogenic CO2" arises from cattle production.

The four-color map theorem lies in the realm of mathematics, not science.

Something was bothered me with this report. How do cows produce anyCO2 (that doesn't come from plant material)? Wrong. The underlying report is fraught with bad science and bad accounting. To get at the figure that cattle are responsible for "9% of all anthropogenic CO2", the authors of "Livestock's Long Shadow" used some potentially dubious accounting, such as double-booking of CO2 emissions to arrive at the statement "[t]he livestock sector accounts for 9 percent of anthropogenic CO2 emissions". Cows don't eat coal or drink gasoline, so how can respiratory and digestive CO2 emitted by cattle count at all? Answer: Deforestation. "The largest share of this [CO2 emissions figure[ derives from land-use changes -- especially deforestation -- caused by expansion of pastures and arable land for feedcrops." (Aside: Boycott Brazilian and Chinese beef and we're fine?) The accounting of CO2 resulting from deforestation is a bit dubious. The report overestimates CO2 sequestration in forestland and and underestimates CO2 sequestration in pasture and crop lands (for example, it completely ignores tilling plant residues back into croplands because of "data unavailability"). The IPCC accounts for transportation of cattle, along with transportation of all other materials, as transportation. Accounting for it with cattle production is again a form of double-booking. Per the report, it is the other way around. Yes, feed lots are a sign of the evil megacorporate world. The cowboy way is even worse. Grass has very little nutrients compared to feed, and what little there is is hard to obtain. The bovine digestive bacteria have to work overtime (read: produce a lot of methane) to enable a cow to gain weight via grazing. Feed is much easier to digest than grass. Moreover, feed can be improved (e.g., immature forages and legumes) and enhanced (e.g., grinding, adding canola oil) to significantly methane production. Some intensive facilities now sequester greenhouse gases produced by cattle, something that simply cannot be done with grazing cattle. Some critiques: http://www.drovers.com/news_editorial.asp?pgid=717&ed_id=4354 http://www.thebrokeronline.eu/en/articles/Livestock-s-long-shadow Alternative reports: http://www.epa.gov/climatechange/emissions/usinventoryreport.html

I think iNow had it right (or mostly right -- he omitted nitrous oxide). The key concern isn't CO2 so much as methane and nitrous oxide, both of which are much more potent greenhouse gases than is CO2 (and nitrous oxide is far stronger than methane). Grass-fed versus corn-fed: Grass-fed beef (or beefalo, or buffalo) requires grazing land. Lots of grazing land. From http://www.fao.org/ag/magazine/0612sp1.htm Also see this 2009 Scientific Americian article and this 2007 Christian Science Monitor article.

Moment, sans qualifier, is usually a synonym for torque. For example, a pure moment results when the net force is zero but the net torque is non-zero. (Yet another term for this: couple.)

Baby Astronaut is correct. arnolp4, you need to learn to distinguish between hypotheses and axioms. Science does have certain core beliefs, or axiomatic statements. The Higgs boson is a hypothetical particle. It is not axiomatic. The unquestioning beliefs in science are pretty simple. Causality: Things have a cause. Moreover, they have a natural cause. Even if we cannot quite get a handle on cause and effect (e.g. radioactive decay), we still believe that things are subject to Natural Laws. So rather than causality, call this the axiom of Natural Laws. Temporality, or the Arrow of Time: Cause precedes effect. (Modern physics sometimes gets a pass on this axiom). Consistency: The laws of nature are invariant with respect to time. Tomorrow will be pretty much the same as yesterday. Yesterday I dropped a plate. It fell to the floor and broke. If I drop a plate tomorrow I should expect the same thing to happen. In particular, it won't fly to the ceiling. Discernability: The laws of nature (see axiom #1) are discernable by us, at least in some form that is approximately correct. Identity: If a=c and b=c, then a=b. Peano's identity axiom implicitly appears throughout science. Neither of the two key reasoning techniques used in the scientific method, inductive and abductive reasoning, is logically valid. Various philosophers of science from Hume to Popper and Kuhn have tried to address the problems of inductive and abductive reasoning; none have entirely succeeded.

Bingo. I smoked some chicken last night for supper. I checked the ashes this morning. Nuts. No gold. There was just gray ashes, plus some yellowish ashes left by the bark, plus some still-hot coals. (Double nuts! I couldn't toss the ashes on the compost; I had to let them cool some more.) Just because every other time I use my smoker and I only have ashes left, does that necessarily mean I won't find gold the next time? Who knows? Maybe those yellowish ashes will be gold the next time around. To a scientist, the answer is no. The residue from my smoking will always be ashes. Science assumes that the universe is logical, consistent, and follows a fairly small set of fairly simple rules. The non-scientific world on the other hand believes the universe is illogical, inconsistent, and capricious; that a man can transform into a goat. That the universe is logical, consistent, simple, and ultimately understandable is a belief. As I mentioned earlier, it is a very-well justified belief (science works). To think that it is anything but a belief is to fall into the trap that abductive reasoning, the core of the scientific method, is a logically valid inference technique.

Garethm you're reference is a free on-line encyclopedia that doesn't support editing (that I can see). The underlined phrase is misleading at best (a nice way of saying "wrong"). Better written the phrase should be, "... or traveling through deep space (far from any massive body) at constant speed". Wikipedia and hyperphysics have much better descriptions of "weightlessness". See http://en.wikipedia.org/wiki/Weightlessness and http://hyperphysics.phy-astr.gsu.edu/Hbase/mass.html#wtls.

Suppose [math]1+2+4+8+\cdots[/math] exists. Call this sum S: [math]S\equiv\sum_{n=0}^{\infty}2^n[/math] Multiply S by two: [math]2S = 2\sum_{n=0}^{\infty}2^n = \sum_{n=0}^{\infty}2\cdot2^n = \sum_{n=0}^{\infty}2^{n+1} = \sum_{n=1}^{\infty}2^n[/math] The last expression is just the original sum sans the zeroth element: [math]2S=S-1[/math] and thus [math]S=-1[/math]. See http://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_·_·_·

Whether nor not zero is a natural number is a bit fuzzy. This is stated up-front in the wikipedia (http://en.wikipedia.org/wiki/Natural_number) and mathworld (http://mathworld.wolfram.com/NaturalNumber.html) articles on natural numbers.

If you were traveling at a constant speed toward the Moon you would not feel weightless. The spacecraft would have to be thrusting continuously to maintain a constant speed. You would however feel weightless if the spacecraft was in free drift. Wile the International Space Station has to reboost occasionally to overcome the slow loss of altitude due to atmosphere friction, it is in free drift most of the time. During these periods of free drift the astronauts on the Station feel weightless. Yet a quick calculation shows that the gravitational force is only reduced by about 10%. So why don't the astronauts on the Space Station feel like they weigh 90% of what they weigh on the Earth's surface? The answer: You do not feel gravity. You cannot even measure (directly) the force due to gravity.

The sum of all counting numbers, [math]1+2+3+4+\cdots = \sum_{n=1}^{\infty}n[/math] does have meaning. By zeta function regularization (see http://motls.blogspot.com/2007/09/zeta-function-regularization.html, http://en.wikipedia.org/wiki/Zeta_function_regularization, http://math.ucr.edu/home/baez/week126.html, and (PDF) http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf), [math]\sum_{n=1}^{\infty}n = -\frac 1 {12}[/math] Zeta function regularization similarly says [math]1+1+1+1+1+\cdots = \sum_{n=1}^{\infty} 1 = -\frac 1 2[/math] Abel summation yields things like [math]1-1+1-1+1-1+\cdots = \frac 1 2[/math] Some other interesting answers for divergent series: [math]1+2+4+8+\cdots = \sum_{n=0}^{\infty}2^n = -1[/math] [math]1-2+4-8+\cdots = \sum_{n=0}^{\infty}(-2)^n = \frac 1 3[/math] Various tricks for evaluating such sums go all the way back to Euler. That [math]1+2+3+4+\cdots=-1/12[/math] has also been verified experimentally in measurements of the Casimir force.

You can't do that. The reason is that you can reorganize a conditionally convergent series to give any answer you want according to the Riemann series theorem.

Dang. So should I. 40 is close to ideal: old enough to have learned a lot academically and practically, nor so old that having a quick mind is a pleasant memory. I remember having a beige toaster on my desk. Pleasant memories ...