the tree

We wouldn't need to stop the planet if we were heading to go past Alpha Centauri, we'd get caught in an orbit like normal.

Couldn't you just I dunno, burn Venus for fuel or something?

Yes you're right about the Kernel and Range (are you required to write proofs for that or just give the answer?) I'm fairly sure the basis is just the standard basis for R2, but don't hold me to that.

Really, maths puns are the first sine of insanity, and I think we're getting a bit off track here anyway. I'm unsure if it's worthwhile covering the derivation of Euler's Formula (though it's worth taking a look at) considering the OP clearly hasn't had a formal introduction to complex numbers so likely doesn't know much about the analysis behind Taylor Series et cetera.

Some plants might adapt in time, or more likely some currently obscure plants will become common. I'd be interested to see how evolution copes with the rate of change - much more catastrophic events have happened in the past and caused completely new genera to become dominant. With climate change it's pretty fast but not fast on the scale of a massive asteroid ruining everything so I think that we'll see more adaptations within the established species. Migratory birds might have an edge over others since they can, as this has shown, pick another place with a more suited environment.

What the hell? I claimed that non divergent sequences can converge or oscillate, for that statement to be true there obviously only needs to be one case of each.

Let's say I have two sequences that don't diverge to infinity: [imath]a_n = (-1)^n[/imath] [imath]b_n = (\frac{2}{3})^n[/imath] One oscillates, the other converges. QEF.

It's worth watching the video on that article as well, the [acr=Royal Society For the Protection of Brids]RSPB[/acr] spokesperson clarifies that the feeding tables are secondary and it's a change in climate that caused the initial change in this behaviour and that bird tables may have caused further changes in the birds physiology. The really interesting thing is that because of the different distances between Germany and Britain versus Germany and Spain mean that the two groups of Blackcaps now have a different mating season which could allow for greater genetic drift and two different species could eventually emerge (the spokesperson says thousands of years but considering these noticeable changes have happened over decades, could it be quicker than that?) Of course the positive news here is the suggestion that many animals will be able to adapt to climate change and rather than mass extinctions across the board there could be exciting new species evolving to cope with different conditions. The overall population of blackcaps is increasing and now maybe diverging into separate populations so what do you think that might mean for biodiversity in general?

Knowing that a sequence doesn't diverge to infinity just doesn't tell you anything, it might converge to a limit or it might oscillate within certain bounds.

No.

First up, good for you. It's always nice to see someone motivated and wanting to learn something beyond the structure of the education system. Now the first thing that you want to learn is differential calculus, or specifically the technique of differentiation. Given that you're working in a limited time frame I think it would be a more realistic aim to get really good at that than to steam through and learn the ins and outs of integral calculus as well. If you read the GUT thread that DJBruce linked to - then search around for some relevant exercise sheets then you should be well on your way.

No need to show off, we all know that everyone learned this from Scrubs.

doG, three circles? I made it 16, with 3 planes on either side of P and 3 circles in each of them save for the pair of impossible circles. (I would upload some images or something but it turns out the Wifi on trains isn't that great) So I figured that the sum of the squares should be: [math]2 \times \Bigg( \sum_{b \in \{ 2,4,6 \}} \sum_{c \in \{ 5,7,9 \}} \begin{cases} c^2 - b^2 & \mbox{if } c>b \\ 0 & \mbox{o/w} \end{cases} \Bigg\} \Bigg) = 616 [/math] Which is the answer the OP got. So either we're both wrong in (probably) the same way, or the textbook is wrong in a different way. Mr. Skeptic, did you get 596 somehow? If so, how?

So guys, tonight I ran into a tree branch. There was lots of blood and a paramedic was involved, it was not fun. Moral: don't run in the dark.

True. Nonetheless, pick an element of R and systematically go through all elements of the empty set to ensure the statement is true.

Oh, x's and y's the other way around, whatever. Equivalence does sort of carry redundancy with it, when the equivalence is trivial.

You'll have to make your notation clearer. Define which are sets and which are entities, and try to avoid recursive definitions. State your formulae and then their implications, if you give a worked example then use a calculator to avoid calculation errors.

This looks a tad like a spambot, methinks.

But there are infinitely many counter examples to that, it doesn't take much effort to set U to 0 and then find some a,b,c,n such that an+bn<cn.

An equation is sentence with an '=' in it. An identity is equation that is true regardless of the variables involves. An equality is true only because of the constants in that particular instance.

triclino: For 4, just take 1 and set X=sqrt(x) and Y=sqrt(y). Similarly for 5, set X=sqrt(x) and Y=0. All you've done is give specific values to the variables, so it is the same lemma. D H: That's sufficient to prove that a positive real has no more than one positive root. Proving that it has at least one would in fact be much easier: I think I just ran into a brick wall by trying to do both at once, thanks though.

On this note. I was thinking today proving that a positive real number has exactly one positive real square root might not be so trivial. The only proof that I could think of was dependent on ideas from real analysis.

Yeah, generally if your going to cite an article then you should give a source. Who wrote this and what journal does it come from? The theory-ladenness of observation is a very interesting and very real problem, which the case of the guest lecturer is not a bad example but without any assurance that the study was not completely made up, it's not a useful one, at all.

Quite seriously ajb and DH know what they are talking about and it would be a good idea to y'know, listen to him. But while we're here: 9 and 10 are both special cases of [imath]x(a+b)=xa+xb[/imath], which is one of the field axoims anyway, so they are clearly true (which means that you wouldn't bother going through it rigoursly unless you were made to for the sake of an exercise). 6 is nearly how you'd define sqrt in the first place. 4 and 5 are made redudant by 1. 1,2,3,7 and 8 might be worth proving once but really it doesn't matter if you don't.

I would put the 'error' down to a simple mistake, but basically the same false claim is made in the second example. 19.50133+11.14333-20.64463=1.338181, not zero. Although, there's no real indication of what this is meant to show.